A D V A N C E S IN I M A G I f ~ G A N D E L E C T R O N PHYSICS
V O L U M E 113
A D V A N C E S IN I M A G I f ~ G A N D E L E C T R O N PHYSICS
V O L U M E 113
EDITOR-IN-CHIEF
PETER W. H A W K E S CEMES/Laboratoire d' Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France
ASSOCIATE E D I T O R S
BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California
TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom
Advances in
Imaging and Electron Physics EDITED BY
P E T E R W. HAWKES CEMES/Laboratoire d"Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France
V O L U M E 113
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This book is printed on acid-free paper. @ Copyright 9 2000 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher's consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2000 chapters are as shown on the title pages: if no fee code appears on the title page, the copy fee is the same as for current chapters. 1076-5670/00 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press article in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press article is given. A C A D E M I C PRESS A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 52 Jamestown Road, London, NW1 7BY, UK http://www.hbuk.co.uk/ap/ International Standard Serial Number: 1076-5670 International Standard Book Number: 0-12-014755-6 Printed in the United States of America 00 01 02 03 EB 9 8 7 6 5 4
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CONTENTS
CONTRIBUTORS PREFACE .
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FORTHCOMING CONTRIBUTIONS .
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vii ix xi
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The Finite Volume, Finite Element, and Finite Difference Methods as Numerical Methods for Physical Field Problems CLAUDIO MATTIUSSI
I. II. III. IV. V.
Introduction . Foundations . Representations Methods . Conclusions . References . .
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1 5 45 86 140 142
The Principles and Interpretations of Annular Dark-Field Z-Contrast Imaging P. D. NELLIST AND S. J. PENNYCOOK I. II. III. IV. V. VI.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Transverse Incoherence . . . . . . . . . . . . . . . . . . Longitudinal Coherence . . . . . . . . . . . . . . . . T h e U l t i m a t e R e s o l u t i o n a n d the I n f o r m a t i o n L i m i t . . . . Q u a n t i t a t i v e I m a g e P r o c e s s i n g a n d Analysis . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
148 153 170 181 190 197 199 200
Measurement of Magnetic Fields and Domain Structures Using a Photoemission Electron Microscope S. A. NEPIJKO, N. N. SEDOV, AND G. SCHONHENSE I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . II. I m a g i n g of F e r r o m a g n e t i c D o m a i n B o u n d a r i e s in a P E E M in the O p e r a t i o n M o d e W i t h o u t R e s t r i c t i o n of the E l e c t r o n B e a m . III. I m a g i n g of F e r r o m a g n e t i c D o m a i n B o u n d a r i e s in P E E M in the Case of R e s t r i c t i o n of the E l e c t r o n B e a m by C o n t r a s t A p e r t u r e or K n i f e - E d g e . . . . . . . . . . . . . . . . . . . . . . . .
205 206
222
CONTENTS
vi
IV. M a g n e t i c D o m a i n I m a g i n g in X - P E E M
Using Magnetic X-Ray
Circular Dichroism . . . . . . . . . . V. M a g n e t i c D o m a i n I m a g i n g in U V - P E E M Kerr-Effect-Like Contrast . . . . . . . VI. C o n c l u s i o n s . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . References . . . . . . . . . . . . . .
. . . . . . . . . . Using a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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228
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239 242 245 246
Improved Laser Scanning Fluorescence Microscopy by Multiphoton Excitation N. S. WHITE AND R. J. ERRINGTON I. I n t r o d u c t i o n . . . . II. F u t u r e P r o s p e c t s . . . References . . . . . . Index
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249 276 277
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279
CONTRIBUTORS Numbers in parentheses indicate the pages on which the author's contribution begins.
R. J. ERRINGTON(249), Department of Medical Biochemistry, University of Wales College of Medicine, Heath Park, Cardiff, U.K. CF14 4XN CLAUDIO MATTIUSSI (1), Clampco Sistemi-NIRLAB, Padriciano 99, 34012 Trieste, Italy P. D. NELLIST (147), Nanoscale Physics Research Laboratory, School of Physics and Astronomy, The University of Birmingham, Birmingham, B15 2TT, U.K. S. A. NEPIJKO (205), Institute of Physics, Ukrainian Academy of Sciences, Pr. Nauki 46, 252022 Kiev, C.I.S./Ukraine S. J. PENNYCOOK (147), Oak Ridge National Laboratory, Solid State Division, PO Box 2008, Oak Ridge, TN 37831-6030, USA G. SCHONHENSE(205), Institut ffir Physik, Johannes Gutenberg-Universit~it Mainz, Staudingerweg 7, 55099 Mainz, F.R.G. N. N. SEDOV(205), The Moscow Military Institute, Golovachev str., 109380 Moscow, C.I.S./Russia N. S. WHITE (249), Bio-Rad Biological Microscopy Unit (BMU), Department of Plant Sciences, University of Oxford, South Parks Road, Oxford, U.K. OX1 3RB
vii
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PREFACE
The four contributions to this volume are concerned, first with a family of numerical methods of broad application but of particular interest in electron imaging and second, with three ways of extracting information from an image-forming instrument. A superficial glance at any text on calculating field distributions will reveal that, among the methods most frequently used are the finite-difference and finite-element methods of transforming a differential equation into a difference equation. To these must be added the finite-volume approach. These methods are of course of immediate interest in numerous other areas than the calculation of electric and magnetic field distributions and the opening chapter, by C. Mattiussi, places this collection of methods in context and brings out very clearly the family resemblance between them. The authors of the second chapter will need no introduction to electron microscopists, for the work of P. D. Nellist and S. J. Pennycook on the interpretation of the signal collected by a wide-angle annular detector in a scanning transmission electron micoroscope is well known. I am delighted that they have agreed to write a connected account of this very exciting technique for these Advances and I have no doubt that this presentation will be found extremely valuable. The photoemission electron microscope is admittedly among the more specialized and least widely known electron instruments but there is no doubt that, for certain types of specimen at least, it is capable of furnishing very useful information. The contribution by S. A. Nepijko, N. N. Sedov and G. Sch6nhense, the first two of whom are well known to the Russian microscopy community, will, I hope, remedy our ignorance of the technique. We conclude with a relatively short contribution on some very important new developments in scanning fluorescence microscopy. I hope to include further and more detailed chapters in the series on this type of imagery and I am delighted that this first contribution already gives a vivid picture of the kind of new results that we can expect. In conclusion, I thank very sincerely all the authors for the trouble they have taken to make their chapters accessible to a wide readership and I list forthcoming contributions. Peter Hawkes
ix
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FORTHCOMING CONTRIBUTIONS M. An and R. Tolimieri
Chromotomographic hyperspectral imaging D. Antzoulatos
Use of the hypermatrix N. Bonnet (vol. 114)
Artificial intelligence and pattern recognition in microscope image processing G. Borgefors Distance transforms A. van den Bos and A. Dekker Resolution P. G. Casazza (vol. 115) Modern tools for Weyl-Heisenberg (Gabor) frame J. A. Dayton Microwave tubes in space E. R. Dougherty and Y. Chen
Granulometries J. M. H. Du Buf
Gabor filters and texture analysis X. Evangelista
Dyadic warped wavelets R. G. Forbes Liquid metal ion sources E. Fi~rster and F.N. Chukhovsky
X-ray optics A. Fox The critical-voltage effect M. I. Herrera
The development of electron microscopy in Spain xi
xii
FORTHCOMING CONTRIBUTIONS
K. lshizuka Contrast transfer and crystal images
C. Jeffries Conservation laws in electromagnetics I. P. Jones ALCHEMI M. Jourlin and J.-C. Pinoli (voi. 115) Logarithmic image processing
E. Kasper Numerical methods in particle optics A. Khursheed (vol. 115) Scanning electron microscope design G. K6gel Positron microscopy W. Krakow Sideband imaging A. van de Laak-Tijssen and T. Muivey (vol. 115) Memoir of J. B. Le Poole J. C. MeGowan Magnetic transfer imaging S. Mikoshiba and F. L. Curzon Plasma displays
E. Oestersehulze Scanning tunnelling microscopy M. A. O'Keefe Electron image simulation B. Olstad Representation of image operators J. C. Paredes and G. R. Aree Stack filtering and smoothing C. Passow Geometric methods of treating energy transport phenomena E. Petajan and F. A. Ponce Nitride semiconductors for high-brightness blue and green light emission
FORTHCOMING CONTRIBUTIONS
,I. W. Rabalais
Scattering and recoil imaging and spectrometry H. Rauch
The wave-particle dualism G. Schmahl X-ray microscopy ,I. P. F. Sellschop Accelerator mass spectroscopy S. Shirai
CRT gun design methods T. Soma Focus-deflection systems and their applications I. Talmon Study of complex fluids by transmission electron microscopy A. Tonazzini and L. Bedini
Image restoration J. Toulouse New developments in ferroelectrics T. Tsutsui and Z. Dechun
Organic electroluminescence, materials and devices Y. Uchikawa Electron gun optics D. van Dyck Very high resolution electron microscopy L. Vincent
Morphology on graphs C. D. Wright and E. W. Hill
Magnetic force microscopy T. Yang (vol. 114) Cellular Neural Networks
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ADVANCESIN IMAGINGAND ELECTRON PHYSICS,VOL. 113
The Finite Volume, Finite Element, and Finite Difference Methods as Numerical Methods for Physical Field Problems Claudio
Mattiussi
Clampco Sistemi-NIRLAB, Area Science Park, Padriciano 99, 34012 Trieste, Italy ~
1
I. Introduction . . . . . . . . . . . . . . . . . . . II. Foundations . . . . . . . . . . . . . . . . . . . A. The Mathematical Structure of Physical Field Theories B. Geometric Objects and Orientation . . C. Physical Laws and Physical Quantities D. Classification of Physical Quantities . E. Topological Laws . . . . . . . . F. Constitutive Relations . . . . . . G. Boundary Conditions and Sources H. The Scope of the Structural Approach III. Representations . . . . . . . . A. Geometry . . . . . . . . B. Fields . . . . . . . . . C. Topological Laws .... D. Constitutive Relations . . . . . . E. Continuous Representations IV. Methods . . . . . . . . . A. The Reference Discretization Strategy B. Finite Difference Methods . . . . . C. Finite Volume Methods . . . . . . D. Finite Element Methods . . . . . . V. Conclusions . . . . . . . . . . . . References . . . . . . . . . . . . .
5 5 7 15 22 27 33 37 38 45 45 55 63 69 72 86 86 112 122 131 140 142
I. INTRODUCTION One of the fundamental --naively
speaking--of
p h y s i c s is t h a t o f f i e l d , i.e.,
concepts of mathematical a spatial distribution
of some mathematical
object
r e p r e s e n t i n g a p h y s i c a l q u a n t i t y . T h e p o w e r o f t h i s i d e a lies in t h a t it a l l o w s t h e m o d e l i n g of a n u m b e r of very i m p o r t a n t p h e n o m e n a , for example, those g r o u p e d under the labels "electromagnetism,"
"thermal conduction," "fluid dynamics,"
"solid mechanics" -- to name a few--and Using the concept transforms
a physical
of the combinations
o f field, a set o f " t r a n s l a t i o n problem
belonging
to
one
thereof.
r u l e s " is d e v i s e d , w h i c h of the
aforementioned
1E_mail
[email protected]
1 Volume 113 ISBN 0-12-014755-6
ADVANCESIN IMAGINGAND ELECTRON PHYSICSCopyright 9 2000by AcademicPress All rights of reproduction in any form reserved. ISSN 1076-5670/00$35.00
CLAUDIO MATTIUSSI
d o m a i n s - - a physical field problem--into a mathematical one. The properties of this mathematical model of the physical p r o b l e m - - a model that usually takes the form of a set of partial differential or integro-differential equations, supplemented by a set of initial and b o u n d a r y c o n d i t i o n s - - c a n then be subjected to analysis in order to establish if the mathematical problem is well posed [Gustafsson et al., 1995]. If the result of this inquiry is judged satisfactory, it is possible to proceed to the actual derivation of the solution, usually with the aid of a computer. The recourse to a computer implies, however, a further step after the modelin9 step described so far, namely the reformulation of the problem in discrete terms, as a finite set of algebraic equations, which are more suitable than a set of partial differential equations to the number-crunching capabilities of present-day computing machines. If this discretization step is made starting from the mathematical problem in terms of partial differential equations, the resulting procedures can be logically called numerical methods for partial differential equations. This is indeed how the finite difference (FD), finite element (FE), finite volume (FV), and many other methods are often categorized. Finally, the system of algebraic equations produced by the discretization step is solved, and the result is interpreted from the point of view of the original physical problem. More than 30 years ago, while considering the impact of the digital computer on mathematical activity, Bellman [1968] wrote Much of the mathematical analysis that was developed over the eighteenth and nineteenth centuries originated in attempts to circumvent arithmetic. With our ability to do large-scale arithmetic ...we can employ simple, direct methods requiring much less old-fashioned mathematical training... This situation by no mean implies that the mathematician has been dispossessed in mathematical physics. It does signify that he is urgently needed.., to transform the original mathematical problems to the stage where a computer can be utilized profitably by someone with a suitable scientific training. ... Good mathematics, like politics, is the art of the possible. Unfortunately, people quickly forget the origins of a mathematical formulation with the result that it soon acquires a life of its own. Its genealogy then protects it from scrutiny. Because the digital computer has so greatly increased our ability to do arithmetic, it is now imperative that we reexamine all the classical mathematical models of mathematical physics from the standpoints of both physical significance and feasibility of numerical solution. It may well turn out that more realistic descriptions are easier to handle conceptually and computationally with the aid of the computer... In this spirit, the present work describes an alternative to the classical partial differential equations-based approach to the discretization of physi-
N U M E R I C A L M E T H O D S F O R PHYSICAL F I E L D P R O B L E M S
3
cal field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possible the physical and geometrical content of the original problem, and is made possible by the existence and properties of a common mathematical structure of physical field theories [Tonti, 1975]. The goal is to maintain the focus, both in the modeling and in the discretization step, on the physics of the problem, thinking in terms of numerical methods for physical field problems, and not for a particular mathematical form (for example, a partial differential equation) into which the original physical problem happens to be translated (Figure 1). The advantages of this approach are various. First of all, it provides a unifying viewpoint for the discretization of physical field problems, which is valid for a multiplicity of theories. Secondly, by basing the discretization of the problems on the structural properties of the theory to which they belong, this approach gives discrete formulations that preserve many physically significant properties of the original problem. Finally, being based on very intuitive geometrical and physical concepts, this approach facilitates both the analysis of existing numerical methods and the development of new ones. The present work considers both these aspects, introducing first a reference discretization strategy directly inspired to the results of the analysis of the structure of physical field theories. Then, a number of popular numerical methods for partial differential equations are considered, and their workings are compared with those of the reference strategy, in order to ascertain to what extent these methods can be interpreted as discretization methods for physical field problems. The realization of this plan requires the preliminary introduction of the basic ideas of the structural analysis of physical field theories. These ideas are very simple, but unfortunately they were formalized and given physically unintuitive names at the time of their first application, within certain branches of advanced mathematics. Therefore, in applying them to other fields, one is faced with the dilemma of inventing for these concepts new and, hopefully, more meaningful names, or maintaining the ones inherited from mathematical tradition. After some hesitation, I chose to keep to the old names, to avoid a proliferation of typically ephemeral new definitions and in consideration of the fact that there can be difficult Concepts, not difficult names; we must try to clarify the former, not avoid the latter [Dolcher, 1978]. The intended audience for this paper is rather wide. The novice to the field of numerical methods for physical field problems will find herein a framework that helps to intuitively grasp the common concepts hidden under the surface of a variety of methods, thus smoothing the path to their mastery. On the other hand, the ideas presented should prove helpful also to the
C L A U D I O MATTIUSSI
physical
"discrete" #
field problem
modelingl
-•m'
continuous" odeling
continuous mathematical mode]: p.d.e.
partially discrete mathematical model
discretization
discretization system of algebraic equations
numerical solution discrete solution I
approx, reconstruction I continuous field representation FIGURE 1. The alternative paths leading from a physical field problem to a system of algebraic equations.
experienced numerical practitioner and to the researcher as additional tools that can be applied to the evaluation of existing methods and the development of new ones. Be that as it may, I will be satisfied if the following pages prove themselves capable of conveying the spirit with which they were conceived--the joy that derives from seeing a well-known subject in a new light. A joy that this contribution represents an attempt at sharing. Finally, it is worth remembering that the result of discretization must be subjected to analysis also, in order to establish its properties as a new
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
5
mathematical problem, and to measure the effects of discretization on the solution when compared to that of nondiscrete mathematical models. This further analysis will not be dealt with here; the emphasis being on the unveiling of the c o m m o n discretization'substratum for existing methods, the convergence, stability, consistency, and error analyses of which a b o u n d in the literature. II. FOUNDATIONS
A. The Mathematical Structure of Physical Field Theories It was mentioned in the Introduction, that the approach to the discretization that will be presented in this work is based on the observation that physical field theories possess a c o m m o n structure. Let us, therefore, start by explaining what we mean when we talk of the structure of a physical theory. It is a c o m m o n experience that the exposure to more than one physical field theory (for example, thermal conduction and electrostatics) aids the comprehension of each single one and facilitates the quick grasping of new ones. This occurs because there are easily recognizable similarities in the mathematical formulation of theories describing different phenomena, which permit the transfer to unfamiliar realms of intuition and imageries developed for more familiar cases. 2 Building in a systematic way on these similarities, one can fill a correspondence table that relates physical quantities and laws playing a similar role within different theories. Usually we say that there are analogies between these theories. These analogies are often reported as a trivial, albeit useful curiosity, but some scholars have devoted considerable efforts to the unveiling of their origin and meaning. In their quest, they have discovered that those similarities can be traced back to the c o m m o n geometric background upon which the "physics" is built. In the b o o k that, building on a long tradition, took those enquiries almost to their present state, Tonti [1975] emphasized: 9 the existence within physical theories of a natural association of m a n y physical quantities, with geometric objects in space and space-time. 3 9 the necessity to consider as oriented, the geometric objects to which physical quantities are associated. 2One may say that this is indeed the essence of explanation, i.e., the mapping of the unexplained on something that is considered obvious. 3For the time being, we give the concept of oriented 9eometric object an intuitive meaning (points, and sufficiently regular lines, surfaces, volumes, and hypervolumes, along with time instants and time intervals).
CLAUDIO MATTIUSSI
9 the existence of two kinds of orientation for these geometric objects. 9 the primacy and priority, in the foundation of each theory, of global physical quantities associated with geometric objects, over the corresponding densities. From this set of observations there follows naturally a classification of physical quantities, based on the type and kind of orientation of the geometric object with which they are associated. The next step is the consideration of the relations held between physical quantities within each theory. Let's call them generically the physical laws. From our point of view, the fundamental observation in this context relates to: 9 the existence within each theory of a set of intrinsically discrete physical laws. These observations can be given a graphical representation as follows. A classification diagram for physical quantities is devised, with a series of "slots" for the housing of physical quantities, each slot corresponding to a different kind of oriented geometric object (Figures 7 and 8). The slots of this diagram can be filled for a number of different theories. Physical laws will be represented in this diagram as links between the slots housing the physical quantities (Figure 17). The classification diagram of physical quantities, complemented by the links representing physical laws, will be called the factorization diagram of the physical field problem, to emphasize its role in singling out the terms in the governing equations of a problem, according to their mathematical and physical properties. The classification and factorization diagrams will be used extensively in this work. They seem to have been first introduced by Roth (see the discussion in Bowden [-1990], who calls them Roth's diagram). Branin E1966] used a modified version of Roth's diagrams, calling them transformation diagrams. Tonti [1975; 1976a; 1976b; 1998] refined and used these diagrams--that he called classification schemes--as the basic representation tool for analysis of the formal structure of physical theories. We will refer here to this last version of the diagrams, which were subsequently adopted by many authors with slight graphical variations and under various names [Oden and Reddy, 1983; Baldomir and Hammond, 1996; Bossavit, 1998a], and for which the name Tonti diagrams was suggested. The Tonti classification and factorization diagrams are an ideal starting point for the discretization of a field problem. The association of physical quantities with geometric objects gives a rationale for the construction of the discretization meshes and the association of the variables to the constituents of the meshes, whereas singling out in the diagram the intrinsi-
N U M E R I C A L M E T H O D S F O R PHYSICAL F I E L D P R O B L E M S
7
cally discrete terms of the field equation permits us both to pursue the direct discrete rendering of these terms and to focus on the discretization effort with the remaining terms. Having found this common starting point for the discretization of field problems, one might be tempted to adopt a very abstract viewpoint, based on a generic field theory, with a corresponding generic terminology and factorization diagram. However, although many problems share the same structure of the diagram, there are classes of theories whose diagrams differ markedly and consequently a generic diagram would have been either too simple to encompass all the cases, or too complicated to work with. For this reason we are going to proceed in concrete terms, selecting a model field theory and referring mainly to it, in the belief that this could aid intuition, even if the reader's main interest is in a different field. Considering the focus of the series where this contribution appears, electromagnetism was selected as the model theory. Readers having another background can easily translate what follows by comparing the factorization diagram for electromagnetism with that of the theory they are interested in. To give a feeling of what is required for the development of the factorization diagram for other theories, the case of heat transfer, thought of as representative of a class of scalar transport equations, is discussed. It must be said that there are still issues that wait to be clarified in relation to the factorization diagrams and the mathematical structure of physical theories. This is true in particular for some issues concerning the position of energy quantities within the diagrams and the role of orientation with reference to time. Luckily this touches only marginally on the application of the theory to the discretization of physical problems finalized to their numerical solution.
B. Geometric Objects and Orientation The concept of geometric object is ubiquitous in physical field theories. For example, in the theory of thermal conduction the heat balance equation links the difference between the amount of heat contained inside a volume V at the initial and final time instants T~ and TI of a time interval I, to the heat flowing through the surface S, which is the boundary of V, and to the heat produced or absorbed within the volume during the time interval. Here, V and S are geometric objects in space, whereas 1, T~ and TI are geometric objects in time. The combination of a space and a time object (e.g., the surface S considered during the time interval I, or the volume V at the time instant T~, or TI) gives a space-time geometric object. These examples show
CLAUDIO MATTIUSSI
FIGURE 2. Internal and external orientation for surfaces.
that by "geometric object" we mean the points and the sufficiently wellbehaved lines, surfaces, volumes, and hypervolumes contained in the domain of the problem, and their combination with time instants and time intervals. This somewhat vague definition will be substituted later by the more detailed concept of p-dimensional cell. The example above also shows that each mention of an object comes with a reference to its orientation. To write the heat balance equation, we must specify if the heat flowing out of a volume or that flowing into it are to be considered positive. This corresponds to the selection of a preferred direction through the surface. 4 Once this direction is chosen, the surface is said to have been given external orientation, where the qualifier "external" hints at the fact that the orientation is specified by means of an arrow that does not lie on the surface. Correspondingly, we will call internal orientation of a surface that which is specified by an arrow lying on the surface and that specifies a sense of rotation on it (Figure 2). Note that the idea of internal orientation for surfaces is seldom mentioned in physics, but very common in everyday objects and in mathematics [Schutz, 1980]. For example, a knob that must be rotated counterclockwise to assure a certain effect is usually designed with a suitable curved arrow drawn on its surface and, in plane affine geometry, the ordering of the coordinate axis corresponds to the choice of a sense of rotation on the plane and defines the orientation of the space. In fact, all geometric objects can be endowed with two kinds of orientations but, for historical reasons, almost no mention of this distinction
4Of course it must be possible to assign such a direction consistently, which is true if the geometric object is orientable [Schutz, 1980], as we will always suppose to be the case. Once the selection has been made, the object acquires a new status: As pointed out by Mac Lane [1986]: "A plane with orientation is really not the same object as one without. The plane with an orientation has more s t r u c t u r e - - n a m e l y , the choice of the orientation."
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
9
survives in physics. 5 Since both kinds of orientation are actually needed in physics, we will show how to build the complete orientation apparatus. We will start with internal orientation, using the affine geometry example above as inspiration. An n-dimensional affine space is oriented by fixing an order of the coordinate axis; this, in the three-dimensional case, corresponds to the choice of a screw-sense, or that of a vortex, in the two-dimensional case, to the choice of a sense of rotation on the plane, and in the one-dimensional case to the choice of a sense (an arrow) along the line. These images can be extended to geometric objects. Therefore, the internal orientation of a volume is given by a screw-sense, that of a surface by a sense of rotation on it, and that of a line by a sense along it (Figure 5). Before proceeding further, it is instructive to consider an example of a physical quantity that, contrary to the c o m m o n belief, is associated with internally oriented surfaces: the magnetic flux ~b. This association is a consequence of the invariance requirement of Maxwell's equations for improper coordinate transformations, that is, those that invert the orientation of space, transforming a right-handed reference system into a lefthanded one. Imagine an experimental setup to probe Faraday's law, for example, verifying the link between the magnetic flux ~b "through" a disk S and the circulation ~ of the electric field intensity E around the loop F, which is the border of S. If we suppose, as is usually the case, that the sign of ~b is determined by a direction through the disk, and that of ~ by the choice of a sense around the loop, a mirror reflection through a plane parallel to the disk axis changes the sign of ~ but not that of ~b. Usually the incongruence is avoided using the right-hand rule to define B and invoking for it the status of axial vector [Jackson, 1975]. In other words, we are told that for space reflections, the sense of the "arrow" of the B vector does not count; only the right-hand rule does. It is, however, apparent that for the invariance of Faraday's law to hold true without such tricks, all we have to do is either to associate 4) with internally oriented surfaces and ~ with internally oriented lines, or to associate ~b with externally oriented surfaces and 0//with lines oriented by a sense of rotation around them (i.e., externally oriented lines, as will be soon clear). Since the effects of an electric field act along the field lines and not around them, the first option seems preferable [Schouten, 1989] (Figure 3). This example shows that the need for the right-hand rule is a consequence of our disregarding the existence of two kinds of orientation. This attitude seems reasonable in physics as we have become used to it in the course of 5But, for example, Maxwell was well aware of the necessity within the context electromagnetism of at least four kinds of mathematical entities for the correct representation of the electromagnetic field (entities referred to lines or to surfaces and endowed with internal or with external orientation) [Maxwell, 1871].
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FIGURE 3. Orientational issues in Faraday's law. The intervention of the right-hand rule, required in the classical version, can be avoided endowing both geometric objects F and S with the same kind of orientation.
our education, but consider that if applied systematically to everyday objects, it would force us to glue an arrow pointing outwards from the above mentioned knob, and to accompany it with a description of the right-hand rule. Note also that the difficulties in the classical formulation of Faraday's law stem from the impossibility to compare directly the orientation of the surface with that of its boundary, when the surface is externally oriented and the bounding line is internally oriented. Here "directly" means "without recourse to the right-hand rule" or similar tricks. The possibility to make this direct comparison is indeed fundamental for the correct statement of many physical laws. This comparison is based on the idea of an orientation induced by an object on its boundary. For example, the sense of rotation that internally orients a surface induces a sense of rotation on its bounding curve, which can be compared with the sense of rotation that orients the surface internally. The same is true for the internal orientation of volumes and of their bounding surfaces. The reader can check that the direct comparison is indeed possible if the object and its boundary are both endowed with internal orientation as defined above for volumes, surfaces, and lines. This raises, however, an interesting issue, since our list of internally oriented objects does not so far include points that nevertheless form the boundary of a line. To make inner orientation a coherent system, we must, therefore, define internal orientations for points (as in algebra we extend the definition of the n-th power of a number to include the case n - 0). This can be done by means of a pair of symbols meaning "inwards" and "outwards" (for example, defining the point a sink or a source, or drawing arrows pointing inwards or outwards) for these images are directly comparable with the internal orientation of a line that starts or ends with the point (Figure 4).
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FIGURE 4. Each internally oriented geometric object induces an internal orientation on the objects that constitute its boundary.
This completes our definition of internal orientation for geometric objects in three-dimensional space, which we will indicate with P, L, S, and V. Let us now tackle the definition of external orientation for the same objects. We said before that in three-dimensional space the external orientation of a surface is given, specifying what turned out to be the internal orientation of a line that does not lie on the surface. This is a particular case of the very definition of external orientation: in an n-dimensional space, the external orientation of a p-dimensional object is specified by the internal orientation of a dual (n-p)-dimensional geometric object [Schouten, 1989]. Hence, in three-dimensional space, external orientation for a volume is specified by an inwards or outwards symbol; for a surface, it is specified by a direction through it; for a line, by a sense of rotation around it; for a point, by the choice of a screw-sense. To distinguish internally oriented objects from externally oriented ones, we will add a tilde to the latter, thus writing P, L, S, and V for externally oriented points, lines, surfaces, and volumes, respectively (Figure 5).
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FIGURE 5. Internal and external orientation for geometric objects in three-dimensional space. The disposition of objects reflects the pairing of reciprocally dual geometric objects.
The definition of external orientation in terms of internal orientation has many consequences. First, contrary to internal orientation, which is a combinatorial concept 6 and does not change when the dimension of the embedding space varies, external orientation depends on it. For example, external orientation for a line in two-dimensional space is assigned by a 6For example, a line can be internally oriented selecting a permutation class (an ordering) of two distinct points on it, which become three nonaligned points for a surface, four noncoplanar points for a volume, and so on.
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FIGURE 6. Each externally oriented geometric object induces an external orientation on the objects that constitute its boundary.
direction through it and not around it as in three-space. 7 Another consequence is the inheritance from internal orientation of the possibility to compare the orientation of an object with that of its boundary, when both are endowed with external orientation. This implies once again the concept of induced orientation, applied in this case to externally oriented objects (Figure 6). The duality of internal and external orientations gives rise to another important pairing, that between dual geometric objects (Figure 5). Note that also in this case the orientation of the objects paired by the duality can be directly compared. However, here the objects have different kinds of orientation. In the context of the mathematical structure of physical theories, this duality plays an important role; for example, it is used in 7Note, however, that the former can be considered the "projection" onto the surface of the latter.
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FIGURE 7. The Tonti classification diagram of physical quantities in three-dimensional space. Each slot is referred to an oriented geometric object, that is, points P, lines L, surfaces S, and volumes V. The left column is devoted to internally oriented objects, and the right column to externally oriented ones. The slots are paired horizontally so as to reflect the duality of the corresponding objects.
the definition of energy quantities and it accounts for some adjointness relationships. We have now at our disposal all the elements required for the construction of a first v e r s i o n - - r e f e r r i n g to the objects of three-dimensional s p a c e - - o f the classification diagram of physical quantities. As anticipated, it consists of a series of slots for the housing of physical quantities, each slot corresponding to an oriented geometric object. To represent graphically the distinction between internal and external orientations, the slots of the diagram are subdivided between two columns. To reflect the important relationship represented by duality, these two c o l u m n s - - f o r internal and external orientations r e s p e c t i v e l y - - a r e reversed with respect to one other, thus making dual objects row-adjacent (Figure 7).
1. Space-Time Objects In the heat balance example that opens this section, it was shown how geometric objects in space, time, and space-time make their appearance in
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the foundation of a physical theory. Until now, we have focused on objects in space and in time; let us extend our analysis to space-time objects. If we adopt a strict space-time viewpoint, that is, if we consider space and time as one, and our objects as p-dimensional objects in a generic fourdimensional space, the extension from space to space-time requires only the application to the four-dimensional case of the definitions given above for oriented geometric objects. However, one cannot deny that in all practical cases (i.e., if a reference frame has to be meaningful for an actual observer) the time coordinate is clearly distinguishable from the spatial ones. Therefore, it seems advisable to consider, in addition to space-time objects per se, the space-time objects considered as cartesian products of a space objects by a time object. Let us list those products. Time can house zero- and one-dimensional geometric objects: time instants T and time intervals I. We can combine these time objects with the four space objects: points P, lines L, surfaces S, and volumes V. We obtain thus eight combinations that, considering the two kind of orientations they can be endowed with, give rise to the sixteen slots of the space-time classification diagram of physical quantities [Tonti, 1976b] (Figure 8). Note that the eight combinations correspond, in fact, to five space-time geometric objects (e.g., a space-time volume can be obtained as a volume in space considered at a time instant, i.e., as the topological product V • T, or as a surface in space considered during a time interval, which corresponds to S • I). This is reflected within the diagram by the sharing of a single slot by the combinations corresponding to the same oriented space-time object. To distinguish space-time objects from merely spatial ones, we will write the former as ~, 5 ~ 5P, ~U, ~ and the latter simply as P, L, S, V. As usual, a tilde signals external orientation.
C. Physical Laws and Physical Quantities In the previous sections, we have implicitly defined a physical quantity (the heat content, the heat flow, and the heat production, in the heat transfer example) as an entity appearing within a physical field theory, which is associated with one (and only one) kind of oriented geometric object. Strictly speaking, the individuation within a physical theory of the actual physical quantities and the attribution of the correct association with oriented geometric objects should be based on an analysis of the formal properties of the mathematical entities that appear in the theory, for example, considering the dimensional properties of those entities and their behavior with respect to coordinate transformations. Given that formal analyses of this kind are available in the literature [Schouten, 1989; Truesdell and Toupin, 1960; Post, 1997-1, the approach within the present work will be more relaxed. To fill in the classification diagram of the
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FIGURE 8. The Tonti space-time classification diagram of physical quantities. Each slot is referred to an oriented space-time geometric object, which is thought of as obtained in terms of a product of an object in space by an object in time. The space objects are those of Figure 7. The time objects are time instants T, and time intervals I. This diagram can be redrawn with the slots referring to generic space-time geometric objects, that is, points ~, lines 5r surfaces ,~, volumes ~/, and hypervolumes , ~ (see Figure 11).
physical quantities of a theory, we will look first at the integrals that appear within the theory, focusing our attention on the integration domains in space and time. This will give us a hint about the geometric object a quantity is associated with. The attribution of orientation to these objects will be based on heuristic considerations deriving from the following fundamental property: The sign of a global quantity associated with a geometric object changes when the orientation of the object is inverted. Further hints would be drawn from physical effects and the presence of the right-hand rule in the traditional definition of a quantity, as well as from the global coherence of the orientation system thus defined. The reader can find in Tonti [1975] an analysis based on a similar rationale, applied to a large
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number of theories, accompanied by the corresponding classification and factorization diagrams.
1. Local and Global Quantities By their very definition, our physical quantities are global quantities, for they are associated with macroscopic space-time domains. This complies with the fact that actual field measurements are always performed on domains having finite extension. When local quantities--densities and r a t e s - - c a n be defined, it is natural to make them inherit the association with the oriented geometric object of the corresponding global quantity. It is, however, apparent that the familiar tools of vector analysis do not allow this association to be represented. This causes a loss of information in the transition from the global to the local representation, when ordinary scalars and vectors are used. For example, from the representation of magnetic flux density with the vector field B, no hint at internally oriented surfaces can be obtained, nor can an association to externally oriented volumes be derived from the representation of charge density with the scalar field p. Usually the association with geometric objects (but not the distinction between internal and external orientations) is reinserted while writing integral relations by means of the "differential term," writing, for example, s B'ds
(1)
and
f
v p dv
(2)
However, given the presence of the integration domains S and V, which accompanies the integration signs, the terms ds a n d d v look redundant. It would be better to use a mathematical representation that refers directly to the oriented geometric object a quantity is associated with. Such a representation exists within the formalism of ordinary and twisted differential forms [de Rham, 1931; Burke, 19853. Within this formalism, the vector field B becomes an ordinary 2-form b 2 and the scalar field p a twisted 3-form/5 3 as follows:
B => b 2
(3)
p ~/5 3
(4)
The symbols b 2 and/5 3 explicitly refer to the fact that magnetic induction and charge density are associated with (and can be integrated only on)
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internally oriented two-dimensional domains and externally oriented threedimensional domains, respectively. Thus, everything seems to conspire for an early adoption of a representation in terms of differential forms. We prefer, however, to delay this step in order to show first how the continuous representation tool they represent can be founded on discrete concepts. Waiting for the suitable discrete concepts to be available, we will temporarily stick to the classical tools of vector calculus. In the meantime, the only concession to the differential form spirit will be the systematic dropping of the "differential" under the integral sign, writing, for example,
fs B
(5)
fpp
(6)
and
instead of Equations (1) and (2).
2. Equations After the introduction of the concept of oriented geometric objects, the next step would ideally be the discussion of the association with them of the physical quantities of the field theory, in our case, electromagnetism. This would parallel the typical development of physical theories, where the discovery of quantities upon which the phenomena of the theory may be conceived to depend precedes the development of the mathematical relations that links those quantities in the theory [Maxwell, 1871]. It turns out, however, that the establishment of the association between physical quantities and geometric objects is based on the analysis of the equations appearing in the theory itself. In particular, it is expedient to list all pertinent equations for the problem considered, and isolate a subset of them, which represent physical laws lending themselves naturally to a discrete rendering, for these clearly expose the correct association. We start, therefore, by listing the equations of electromagnetism. We will first give a local rendition of all the equations, even of those that will eventually turn out to have an intrinsically global nature, since this is the form that is typically considered in mathematical physics. The first pair of electromagnetic equations that we consider represent in local form Gauss's law for magnetic flux [Equation (7)] and Faraday's induction law [Equation (8)] div B = 0 (7) c~B curl E + -~- - 0
(8)
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where B is the magnetic flux density and E is the electric field intensity. We will show below that these equations have a counterpart in the law of charge conservation 0p div J + --~ = 0
(9)
where J is the electric current density and p is the electric charge density. Similarly, Equations (10) and (11), which define the scalar potential V and vector potential A, curl A = B
(10)
0A -grad V--= E c~t
(11)
are paralleled by Gauss's law of electrostatics [Equation (12)] and Maxwell-Amp6re's law [Equation (13)]--where D is the electric flux density and H is the magnetic field intensity--which close the list of differential statements. div D = p 0D
curl H
= J
(12) (13)
We finally have a list of constitutive equations. A very general form, accounting for most material behaviors, is D(r, t) -
f';o j'fo
F~(E, r', "c)
(14)
F. (H, r', z)
(15)
F~ (E, r', r)
(16)
o
B(r, t) =
to
J (r, t) = o
but, typically, the purely local relations D(r, t) - f~ (E, r, t)
(17)
B(r, t) - fu (H, r, t)
(18)
J (r, t) - f~ (E, r, t)
(19)
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or the even simpler relations D(r) = e(r) E(r)
(20)
B(r) = #(r) H(r)
(21)
a(r) = a(r) E(r)
(22)
adequately represent most actual material behaviors. We will now consider all these equations, aiming at their exact rendering in terms of global quantities. Integrating Equations (7) through (13) on suitable spatial domains, writing c~D for the boundary of a domain D, and making use of Gauss's divergence theorem and Stokes's theorem, we obtain the following integral expressions
~vB = fv 0
(23)
E + ~
B =
0
(24)
J + ~
p =
0
(25)
~os
f, A=fB ~
v-
~
A =
E
s H -~
(26)
(27)
(28)
D =
a
(29)
Note that in Equations (23), (24), and (25) we have integrated the null term on the right-hand side. This was done in consideration of the fact that the corresponding equations assert the vanishing of some kind of physical quantity, and we must investigate what kind of association it has. Moreover in Equations (25), (28), and (29) we added a tilde to the symbol of the integration domains. These are the domains that will turn out later to have external orientation. In Equations (24), (25), (27), and (29) a time derivative remains. A further integration can be performed on a time interval I = I T 1, T2], to eliminate
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FOR PHYSICAL FIELD PROBLEMS
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this residual derivative. For example, Equation (24) becomes w + T1
S
B
= __ITa
0
(30)
1
We adopt a more compact notation, which uses I for the time interval. Moreover, we will consider as an "integral on time instants," a term evaluated at that instant, according to the following symbolism
Correspondingly, since the initial and final instants of a time interval I are actually the boundary 0I of I, we write boundary terms as follows:
R e m a r k : The boundary of an oriented geometric object is constituted by its faces endowed with the induced orientation (Figures 4 and 6). For the case of a time interval I = [T~, T2], the faces that appear in the boundary 0I correspond to the two time instants T~ and T2. If the time interval I is internally oriented in the direction of increasing time, T~ appears in 0I oriented as a source, whereas T 2 appears in it oriented as a sink. On the other hand, as time instants, T~ and T2 are endowed with a default orientation of their own. Let us assume that the default internal orientation of all time instants is as sinks; it follows that 0I is constituted by T 2 taken with its default orientation and by T~ taken with the opposite of its default orientation. We can express symbolically this fact writing c3I = T 2 - T 1 , where the "minus" sign signals the inversion of the orientation of T1. Correspondingly, if there is a quantity Q associated with the time instants, and Q1 and Q2 a r e associated with T~ and T2, respectively, the quantity Q2 - Q~ will be associated with c3I. These facts will be given a more precise formulation later, using the concepts of chain and cochain. For the time being, this example gives a first idea of the key role played by the concept of orientation of space-time geometric objects, in a number of common mathematical operations such as the increment of a quantity and the fact that an expression like 5Tr2df corresponds to (fiT2--fiT,) and not to its opposite. In this context, we signal to the attention of the reader the fact that if the time axis is externally oriented, it is the time instants that are oriented by means of a (through) direction, whereas the time intervals are oriented as sources or sinks.
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With these definitions [Equations (34) and (32)], Equations (23) through (29) become
frfosA=frfsB
(36)
f~fo D=f~f~p
(38)
fTf,~ H-;~Tf D=f.if J
(39)
The equations in this form can be used to determine the correct association of physical quantities with geometric objects.
D. Classificationof PhysicalQuantities In Equations (33) through (39) above, we can identify a number of recurrent terms and deduce from them an association of physical quantities with geometric objects. From Equations (33) and (34) we get
f,f,E=(LxI) frfsB.(S•
(40)
(41)
where the arrow means "is associated with." The term in Equation (41) confirms the association of magnetic induction with surfaces, and suggests a further one with time instants, whereas Equation (40) shows that the electric field is associated with lines and time intervals. These geometric objects are endowed with internal orientation, as follows from the analysis made above for the orientational issues in Faraday's law.
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The status of electric current and charge as a physical quantity can be deduced from Equation (35), giving the terms J ~ (S x I)
(42)
p ~ (V x r )
(43)
showing that electric current is associated with surfaces and time intervals, whereas charge is associated with volumes and time instants. Since the current is due to a flow of charges through the surface, a natural external orientation for surfaces follows. Given this association of electric current with externally oriented surfaces, the volumes to which charge content is associated must also be externally oriented to permit direct comparison of the sign of the quantities in Equation (35). The same rationale can be applied to the terms appearing in Equations (38) and (39), that is, H ~ (L x I)
(44)
D ~ ( S x r)
(45)
This shows that the magnetic field is associated with lines and time intervals and electric displacement with surfaces and time instants. As for orientation, the magnetic field is traditionally associated with internally oriented lines but this choice requires the right-hand rule to make the comparison, in Equation (39), of the direction of H along c3S with the direction of the current flow through the surface S. Hence, to dispense with the use of the right-hand rule, the magnetic field must be associated with externally oriented lines. The same argument applies in suggesting an external orientation for surfaces to which electric displacement is associated. Finally, Equations (26) and (27) give the terms
fi fP V =*"(P x I)
(46)
frf, A--(LxT)
(47)
which show that the scalar potential is associated with points and time intervals, whereas the vector potential is associated with lines and time instants. From the association of the electric field with internally oriented lines, it follows that for the electromagnetic potentials, the orientation is also internal.
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FIGURE 9. The Tonti classification diagram of local electromagnetic quantities.
The null right-hand-side terms in Equations (33), (34), and (35) remain to be taken into consideration. We will see below that these terms express the vanishing of magnetic flux creation (or the nonexistence of magnetic charge) and the vanishing of electric charge creation, respectively. For the time being, we will simply insert them as zero terms in the appropriate slot of the classification diagram for the physical quantities of electromagnetism, which summarizes the results of our analysis (Figure 9).
1. Space-time Viewpoint The terms ~ ~p p and ~7 ~,I in Equations (42) and (43) refer to the same global physical quantity: electric charge. Moreover, total integration is performed in both cases on externally oriented, three-dimensional domains in space-time. We can, therefore, say that charge is actually associated with externally oriented, three-dimensional space-time domains of which a three-
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space volume considered at a time instant, and a three-space surface considered during a time interval, are particular cases. To distinguish these two embodiments of the charge concept, we use the terms charge content-referring to volumes and time i n s t a n t s - - a n d charge flow--referring to surfaces and time intervals. A similar distinction can be drawn for other quantities. For example, the terms ~I ~L E and ~r Is B in Equations (40) and (41) are both magnetic fluxes associated with two-dimensional space-time domains of which we could say that the electric field refers to a "flow" of magnetic flux tubes that cross internally oriented lines, while magnetic induction refers to a surface "content" of such tubes. Since the term content refers properly to volumes and flow to surfaces, it appears preferable to distinguish the two manifestations of each global quantity using an index derived from the letter traditionally used for the corresponding local quantity, as in
fffp=QO(Px) f'i frs J = Qj(~ x
I)
(48) (49)
and
frfs B = d~b(Sx T)
(50)
f* fL E = ~e(L x I)
(51)
The same argument can be applied to electric flux
f~f~
D = 0a(S x T)
(52)
f~fz
H = 0h(~, X i)
(53)
and to the potentials in global form
f T f lA = q.l~(Lx T) lift V= ~#v(p x I)
(54)
(55)
With these definitions we can fill the classification diagram of global electromagnetic quantities (Figure 10).
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FIGURE 10. The Tonti classification diagram of global electromagnetic quantities.
Note that the classification diagrams (Figures 8, 9, and 10) emphasize the pairing of physical quantities which happen to be the static and dynamic manifestations of a unique space-time entity. We can group these variables under a single heading, obtaining a classification diagram of the space-time global electromagnetic quantities U, ~b, ~, Q (Figure 11), which corresponds to the one that could be drawn for local quantities in four-dimensional notation. Note also that all the global quantities of a column possess the same physical dimension; for example, the terms in Equations (48), (49), (52), and (53) all have the physical dimension of electric charge. Nonetheless, quantities appearing in different rows of a column refer to different physical quantities since, even if the physical dimension is the same, the underlying space-time oriented geometric object is not. This fact is reflected in the relativistic behavior of these quantities. When an observer changes his reference frame, his perception of what is time and what is space changes and with it his method of splitting a given space-time physical quantity into
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FIGURE 11. The Tonti classification diagram of global electromagnetic quantities, referring to generic space-time geometric object.
its two "space plus time" manifestations. Hence the transformation laws, which account for the change of reference frame, will combine only quantities referring to the same space-time oriented object. In a four-dimensional treatment such quantities will be logically grouped within a unique entity (e.g., the charge-current vector, the four-dimensional potential, the first and second electromagnetic t e n s o r - - o r the corresponding differential f o r m s - grouping E and B, and H and D, respectively, and so on).
E. Topological Laws Now that we have seen how to proceed to the individuation and classification of the physical quantities of a theory, there remains, as a last step in the determination of the structure of the theory itself, the establishment of
CLAUDIO MATTIUSSI
28
the links existing between the quantities, accompanied by an analysis of the properties of these links. As anticipated, the main result of this further analysis--valid for all field theories--will be the singling out of a set of physical laws, which lend themselves naturally to a discrete rendering, as opposed to another set of relations, which constitute instead an obstacle to the complete discrete rendering of field problems. It is apparent from the definitions given in Equations (48) through (55) that Equations (33) through (39) can be rewritten in terms of global quantities only, as follows: ~bb(3V x T ) = 0(V x T) ~e(~s
X
I) + dpb(S x c~I) = O(S • I)
QJ(~" x ]') + QO(~ x c~I) = O(V x 7) Ua(ctS x T ) =
dpb(S x T)
--~l~(OL x I) -- dll"(L x 01) = dpe(L x I)
r r
x 7') = Qp(V x 7')
x 7 ) - 0'~(,S x 8 ] ' ) = QJ(S x I)
(56) (57) (58) (59) (60) (61)
(62)
Note that no material parameters appear in these equations, and that the transition from the local, differential statements in Equations (7) through (13) to these global statements was performed without recourse to any approximation. This proves their intrinsic discrete nature. Let us examine and interpret these statements one by one. Gauss's magnetic law [Equation (56)] asserts the vanishing of magnetic flux associated with closed surfaces c~V in space considered at a time instant T. From what we said above about space-time objects, there must be a corresponding assertion for time-like closed surface. Faraday's induction law [Equation (57)] is indeed such an assertion for a cylindrical closed surface in space-time constructed as follows (Figure 12) [Truesdell and Toupin, 1960; Bamberg and Sternberg, 1988]" the surface S at the time instant 7'1 constitutes the first base of a cylinder; the boundary of S, c~S, considered during the time interval I = [T 1, T2], constitutes the lateral surface of the cylinder, which is finally closed by the surface S considered at the time instant T2 (remember that T1 and T2, together, constitute the boundary c~I of the time interval I, hence, the term S x c~I in Equation (57) represents the two bases of the cylinder). This geometrical interpretation of Faraday's law is particularly interesting for numerical applications, for it is an exact statement linking physical quantities at times T< T2 to a quantity defined at time T2. Hence, this statement is a good starting point since the development of the time-stepping procedure.
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FIGURE 12. Faraday'sinduction law admits a geometrical interpretation as a conservation law on a space-time cylinder. The (internal) orientation of geometric objects is not represented.
In summary, Gauss's law and Faraday's induction are the space and the space-time parts, respectively, of a single statement: The magnetic flux associated with the boundary of a space-time volume ~ is always zero. q5(c3~) = 0 ( ~ )
(63)
(Remember that the boundary of an oriented geometric object must always be thought of as endowed with the induced orientation.) Equation (63), also called the law of conservation of magnetic flux [Truesdell and Toupin, 1960], gives to its right-hand-side term the meaning of a zero in the production of magnetic flux. From another point of view, the right-hand side of Equation (56) expresses the nonexistence of magnetic charge, and that of Equation (57) the nonexistence of magnetic charge current. The other conservation statement of electromagnetism is the law of the conservation of electric charge [Equation (58)]. In strict analogy with the geometric interpretation of Faraday's law, a cylindrical, space-time, closed.. hypersurface is constructed as follows: the volume V at the time instant 7"1
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constitutes the first base of a hypercylinder; the boundary of V, 0V, considered during the time interval I = [T 1, T2], constitutes the lateral surface of the hypercylinder, which is finally closed by the volume V considered at the time instant T2. The law of charge conservation asserts the vanishing of the electric charge associated with this closed hypercylinder. This conservation statement can be referred to the boundary of a generic space-time hypervolume ~ , giving the following statement, analogous to Equation (63): Q(c~W) = 0(3r ~)
(64)
In Equation (64) the zero on the right-hand side states the vanishing of the production of electric charge. Note that in this case a purely spatial statement, corresponding to Gauss's law of magnetostatics [Equation (56)], is not given, for in four-dimensional space-time a hypervolume can be obtained only as a product of a volume in space multiplied by a time interval. The two conservation statements [Equations (63) and (64)] can be considered the two cornerstones of electromagnetic theory [Truesdell and Toupin, 1960]. De Rham [1931] proved that from the global validity of statements of this kind (or, if you prefer, of Equations (33), (34), and (35)) in a homologically trivial space, follows the existence of field quantities that can be considered the potentials of the densities of the physical quantities appearing in the global statements. In our case we know that the field quantities Vand A, defined by Equations (10) and (11), are indeed traditionally called the electromagnetic potentials. Correspondingly, the field quantities H and D, defined by Equations (12) and (13), are also potentials, and can be called the charge-current potentials (Truesdell and Toupin, 1960). In fact, the definition of H and D is a consequence of charge conservation, exactly as the definition of V and A is a consequence of magnetic flux conservation, therefore, neither are uniquely defined by the conservation laws of electromagnetism. Only the choice of a gauge, for the electromagnetic potentials, and the hypothesis about the media properties for chargecurrent potentials, removes this nonuniqueness. In any case, the global renditions [Equations (59), (60), (61), and (62)] of the equations defining the potentials prove the intrinsic discrete status of Gauss's law of electrostatics, of Maxwell-Amp~re's law, and of the defining equations of the electromagnetic potentials. A geometric interpretation can be given to these laws, too. Gauss's law of electrostatics asserts the balance of the electric charge contained in a volume with the electric flux through the surface that bounds the volume. Similarly, Maxwell-Amp~re's law defines this balance between the charge contained within a space-time volume and the electric flux through its boundary, which is a cylindrical space-time closed surface analogous to the one appearing in Faraday's law,
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31
FIGURE 13. Maxwell-Amp6re's law admits a geometrical interpretation as a balance law on a space-time cylinder. The (external) orientation of geometric objects is not represented.
but with external orientation (Figure 13). This geometric interpretation, like that of Faraday's law, is instrumental for a correct setup of the time-stepping within a numerical procedure. Equation (59) and (60) can be condensed in a single space-time statement, asserting the balance of the electric charge associated with arbitrary space-time volumes with the electric flux associated with their boundaries 0(c~) = Q(~)
(65)
Analogous interpretations hold for Equations (59) and (60), relative to a balance of magnetic fluxes associated with space-time surfaces and their boundaries ~
= ~b(S)
(66)
We can insert the global space-time statements [Equations (63), (64), (65), and (66)], in the space-time classification diagram of the electromagnetic physical quantities (Figure 14). Note that all these statements appear as
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FIGURE 14. The position of topological laws in the Tonti classification diagram of electromagnetic quantities.
vertical links. These links relate a quantity associated with an oriented geometric object with a quantity associated to the boundary of that object (which has, therefore, the same kind of orientation). What is shown here for the case of electromagnetism applies to the great majority of physical field theories. Typically, a subset of the equations, which form a physical field theory, link a global quantity associated with an oriented geometric object to the global quantity that, within the theory, is associated with the boundary of that object [-Tonti, 1975]. These laws are intrinsically discrete, for they state a balance of these global quantities (or a conservation of them, if one of the terms is zero) whose validity does not depend on metrical or material properties, and is, therefore, invariant for very general transformations. This gives them a "topological significance" [Truesdell and Toupin, 1960], which justifies our calling them topological laws. The significance of this finding for numerical methods is obvious: once the domain of a field
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
33
problem has been suitably discretized, topological laws can be written directly and exactly in discrete form.
F. Constitutive Relations To complete our analysis of the equations of electromagnetism, we must consider the set of constitutive equations, represented, for example, by Equations (14), (15), and (16). We emphasize once again that each instance of these kinds of equations is only a particular case of the various forms that the constitutive links between the problem's quantities can take. In fact, while topological laws can be considered universal laws linking the field quantities of a theory, constitutive relations are merely definitions of ideal materials given within the framework of that particular field theory [Truesdell and Noll, 1965]. In other words, they are abstractions inspired by the observation of the behavior of actual materials. More sophisticated models have terms that account for a wider range of observed material behaviors, such as nonlinearity, anisotropy, nonlocality, hysteresis, and the combinations thereof [Post, 1997]. This added complexity usually implies a greater sophistication of the numerical solvers, but does not change the essence of what we are about to say concerning the discretization of constitutive relations. If we consider the position of constitutive relations in the classification diagram of the physical quantities of electromagnetism, we observe that they constitute a link that connects the two columns (Figure 15). This fact reveals that, unlike topological laws, constitutive relations link quantities associated with geometric objects endowed with different kinds of orientation. F r o m the point of view of numerical methods, the main differences with topological laws are the observation that constitutive relations contain material parameters s and the fact that they are not intrinsically discrete. The presence of a term of this kind in the field equations is not surprising, since o t h e r w i s e - - g i v e n the intrinsic discreteness of topological l a w s - - i t would always be possible to exactly discretize and solve numerically a field problem, and we know that this is not the case. Constitutive relations can be transformed into exact links between global quantities only if the local properties do not vary in the domain where the link must be valid. This means that we must impose a series of uniformity requirements on material Sin some cases material parameters seeminglydisappear from constitutive equations. This is the case, for example, of electromagneticequations in empty space adopting gaussian units and setting c = 1. This induces the temptation to identify physical quantities--in this case E and D, and B and H, respectively. However, the approach based on the association to oriented geometric objects reveal that these quantities have actually a quite distinct nature.
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FIGURE 15. The Tonti factorization diagram of electromagnetism in local form. Topological laws are represented by vertical links within columns, whereas constitutive relations are represented by transverse links bridging the two columns of the diagram.
and field properties for a global statement to hold true. On the contrary, since, aside from discontinuities, these requirements are automatically satisfied in the small, the local statement always applies. The uniformity requirement is in fact the method used to experimentally investigate these laws. For example, we can investigate the constitutive relation D = eE
(67)
examining a capacitor with two-plane parallel plates of area A, having a distance l between them and filled with a uniform, linear, isotropic medium having relative permettivity er. With this assumption, Equation (67)
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35
corresponds approximately to --=
A
V 1
e--
(68)
where ~ is the electric flux and V the voltage between the plates. Note that to write Equation (68), we invoke the concepts of planarity, parallelism, area, distance, and orthogonality, which are not topological concepts. This shows that, unlike topological laws, constitutive relations imply the recourse to metrical concepts. This is not apparent in Equation (67), f o r - - a s explained a b o v e - - t h e use of vectors to represent field quantities tends to hide the geometric details of the theory. Equation (67) written in terms of differential forms, or a geometric representation thereof, reveals the presence, within the link, of the metric tensor [Post, 1997; Burke, 1985]. The local nature of constitutive relations can be interpreted saying that these equations summarize at a macroscopic level something going on at a subjacent scale. This hypothesis may help the intuition, but it is not necessary if we are willing to interpret them as definitions of ideal materials. By so doing, we can avoid the difficulties implicit in the creation of a convincing derivation of field concepts from a corpuscular viewpoint. There are other informations about constitutive equations that can be derived observing their position in the factorization diagram. These are not of direct relevance from a numerical viewpoint, but can help to understand better the nature of each term. For example, it has been observed that when the two columns of the factorization diagram are properly aligned according to duality, constitutive relations linked to irreversible processes (for example, Ohm's law linking E and J in Figure 15) appear as slanted links, whereas those representing reversible processes are not [Tonti, 1975]. 1. Constitutive Equations and Discretization Error
We anticipated above that, from our point of view, the main consequence of the peculiar nature of constitutive relations lies in their preventing, in general, the attainment of an exact discrete solution. By exact discrete solution, we mean the exact solution of the continuous mathematical model (for example, a partial differential equation) into which the physical problem is usually transformed. We hinted in the introduction at the fact that the numerical solution of a field problem implies three phases (Figure 1): 1. the transformation of the physical problem into a mathematical model 2. the discretization of the mathematical model 3. the solution of the system of algebraic equations produced by the discretization
CLAUDIO MATTIUSSI
36
physical field problem
I
modelling
error
mathematical model
discretization error
1 system of algebraic equations
L
solver error
discrete solution FIGURE 16.
The three kinds of error associated with the numerical solution of a field
problem.
(The fourth phase represented in Figure 1, the approximate reconstruction of the field function based on the discrete solution, obviously does not affect the accuracy of the discrete solution.) Correspondingly, there will be three kinds of error (Figure 16) [Ferziger and Peri6, 1996; Lilek and Peri6, 1995]: 1. the modeling error 2. the discretization error 3. the solver error Modeling errors are a consequence of the simplifying assumptions about the phenomena and processes, made during the transition from the physical problem to its mathematical model in terms of equations and boundary conditions. Solver errors are a consequence of the limited numerical preci-
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37
sion and time available for the solution of the system of algebraic equations. Discretization errors act between these two steps, preventing the attainment of the exact discrete solution of the mathematical model, even in the hypothesis that our algebraic solvers were perfect. The existence of discretization errors is a well-known fact, but it is the analysis based on the mathematical structure of physical theories that reveals where the discretization obstacle lies, that is, within constitutive relations, topological laws not implying in themselves any discretization error. As anticipated in the Introduction, this in turn suggests the adoption of a discretization strategy where what is intrinsically discrete is included as such in the model, and the discretization effort is focused on what remains. It must be said, however, that once the discretization error is brought into by the presence of the constitutive terms, it is the joint contribution of the approximation implied by the discretization of these terms and of our enforcing only a finite number of topological relations in place of the infinitely many that are implied by the corresponding physical law, that shapes the actual discretization error. This fact will be examined in detail below.
G. Boundary Conditions and Sources In addition to the field equations, a field problem includes a set of boundary conditions and the specification that certain terms appearing in the equations are assigned as sources. Boundary conditions and sources are a means to limit the scope of the problem actually analyzed, for they summarize the effects of interactions with domains or phenomena that we choose not to consider in detail. Let us see how boundary conditions and sources enter into the framework developed above for the equations, with a classification that parallels the distinction between topological laws and constitutive relations. When boundary conditions and sources are specified as given values of some of the field quantities of the problem, they correspond in our scheme to global values assigned to some geometric object placed along the boundary or lying within the domain. Hence, the corresponding values enter the calculations exactly, but for the possibly limited precision with which they are calculated from the corresponding field functions (usually by numerical integration) when they are not directly given as global quantities. Consequently, in this case these terms can be assimilated with topological prescriptions. In other cases boundary and source terms are assigned in the form of equations linking a problem's field variable to a given excitation. In these
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cases, these terms must be considered as additional constitutive relations to which all the considerations made above for this kind of equations apply. In particular, within a numerical formulation, they must be subjected to a specific discretization process. This is, for example, the case for convective boundary conditions in heat transfer problems. In still other cases boundary conditions summarize the effects on the problem domain of the structure of that part of space-time that lies outside the problem domain. Think, for example, about radiative boundary conditions in electrodynamics, and inlet and outlet boundary conditions in fluid dynamics. In these cases, one cannot give general prescriptions, for the representation depends on the geometric and physical structure of this "outside." Physically speaking, a good approach consists in extending the problem's domain, enclosing it in a (hopefully thin) shell whose properties account with a sufficient approximation the effect of the whole space surrounding the domain, and whose boundary conditions belong to one of the previous kinds. This shell can then be modeled and discretized following the rules used for the rest of the problem's domain. However, the devising of the properties of such a shell is usually not a trivial task. In any case, the point we are trying to make is that boundary conditions and source terms can be brought back to topological laws and constitutive relations by physical reasoning, and from there they require no special treatment with respect to what applies to these two categories of relations.
H. The Scope of the Structural Approach The example of electromagnetism, examined in detail in the previous sections, shows that to approach the numerical solution of a field problem taking into account its mathematical structure, we must first classify the physical quantities appearing in the field equations, according to their association with oriented geometric objects, and then factorize the field equations themselves, to the point of being able to draw the factorization diagram for the field theory to which the problem belongs. The result will be a distinction of topological laws, which are intrinsically discrete, from constitutive relations, which admit only approximate discrete renderings (Figure 17). Let us examine briefly how this process works for other theories, and what difficulties we can expect to encounter. From electromagnetism we can easily derive the diagrams of electrostatics and magnetostatics. Dropping the time dependence, the factorization diagram for electromagnetism splits naturally into the two distinct diagrams of electrostatics and magnetostatics (Figures 18 and 19).
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39
FIGURE 17. The distinction between topological and constitutive terms of the field equations, as it appears in the Tonti factorization diagram. Topological laws appear as vertical links and are intrinsically discrete, whereas constitutive relations appear as transverse links and in general permit only approximate discrete renderings.
Given the well-known analogy between stationary heat conduction and electrostatics [Burnett, 1987; Maxwell, 1884], one would expect to derive the diagram for this last theory directly from that of electrostatics. An analysis of physical quantities, reveals, however, that the analogy is not perfect. Temperature, which is linked by the analogy to electrostatic potential V, is indeed associated, like V, to internally oriented points and time intervals, but heat flow density, traditionally considered analogous with electric displacement D, is in fact associated with externally oriented surfaces and time intervals, whereas D is associated with surfaces and time instants. In the stationary case, this distinction makes little difference, but we will see below (Figure 20) that this results in a slanting of the constitutive link between the temperature gradient g and the diffusive heat flux density qd, whereas the
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FIGURE 18. The Tonti factorization diagram for electrostatics in local form.
constitutive link between E and D is not slanted. This reflects the irreversible nature of the former process, as opposed to the reversible nature of the latter. Since the heat transfer equation can be considered a prototype of all scalar transport equations, it is worth examining in detail, including both the nonstationary and convective terms. A heat transfer equation that is general enough for our purposes can be written as follows [Versteeg and Malalasekera, 1995] O(pcO)
0t
+ div(pcOu) - div(k grad 0) = a
(69)
where 0 is the temperature, p is the mass density, c is the specific heat, u is the fluid velocity, k is the thermal conductivity, and a is the heat production density rate. Note that we always start with field equations written in local
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FIGURE 19. The Tonti factorization diagram for magnetostatics in local form.
form, for usually these equations include constitutive terms. We must first factor out these terms, before we can write the topological terms in their primitive, discrete form. Disentangling the constitutive relations from the topological laws, we obtain the following set of topological equations grad 0 = g
(70)
C3qc 0t F divqu + divqe = a
(71)
and the following set of constitutive equations qu = pcOu
(72)
qd = - k g
(73)
qc = pcO
(74)
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FIGURE 20. The Tonti factorization diagram for the heat transfer equation in local form. Note the presence of terms derived from the diagrams of other theories, or other domains.
To write Equations (70) through (74), we have introduced four new local physical quantities: the temperature gradient g, the diffusive heat flow density qd, the convective heat flow density qu, and the heat content density qc. Note that of the three constitutive equations, Equation (72) appears as a result of a driving source term, with the parameter u derived from an "external" problem. This is an example of how the information about interacting phenomena is carried by terms appearing in the form of constitutive relations. Another example is given by boundary conditions
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43
describing a convective heat exchange through a part c3D~ of the domain boundary. If 0oo is the external ambient temperature, h is the coefficient of convective heat exchange and we denote with q~ and 0~ the convective heat flow density and the temperature at a generic point of c3D~, we can write
qv = h(Ov -- 0oo)
(75)
An alternative approach is to consider this an example of coupled problems, where the phenomena that originate the external driving terms are treated as separate interacting problems, which must be also discretized and solved. In this case, a factorization diagram must be built for each physical field problem intervening in the whole problem, and what is treated here as driving terms become links between the diagrams. In these cases, a preliminary classification of all the physical variables appearing in the different phenomena is required, in order to select the best common discretization substratum, especially for what concerns the geometry. Putting the topological laws, with the new boundary term [Equation (75)], in full integral form, we have
;foLO=flfLg f.ifo~q~+f.ffo
q.+f.i;~qa+fo.ff
qc=fTf~a
(76)
(77)
We can define the following global quantities
f
(7s)
f,f,g=G(LxI)
(79)
, f ~ 0 = e(P • I)
f,f
q. = Q.(S x I)
(80)
q~ = Q~(S x I)
(81)
qa = Qa(S x I)
(82)
q~ = Qc(V x T)
(83)
a
(84)
= F ( V x I) -
-
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with the temperature impulse | associated with internally oriented points and time intervals, the thermal tension G associated with internally oriented lines and time intervals, the convective and diffusive heat flows Qu, Qv, and Qd associated with externally oriented surfaces and time intervals, the heat content Qc associated with externally oriented volumes and time instants, and the heat production F associated with externally oriented volumes and time intervals. The same associations hold for the corresponding local quantities. This permits us to write Equations (76) and (77) in terms of global quantities only x I) = G(L x I)
(85)
Qv(c3I~ x I) + Q,(c31~ x I) + Qn(017 x I) + Qc(I~ x 0 T ) = F(17 x 7)
(86)
|
Note that Equation (86) is the natural candidate for the setup of a time-stepping scheme within a numerical procedure, for it links exactly quantities defined at times, which precede the final instant of the interval I, to the heat content Qc at the final instant. This completes our analysis of the structure of heat transfer problems represented by Equation (69), and establishes the basis for their discretization. The corresponding factorization diagram in terms of local field quantities is depicted in Figure 20. Along similar lines one can conduct the analysis for many other theories. No difficulties are to be expected for those that happen to be characterized--like electromagnetism and heat transfer--by scalar global quantities. More complex appears the case of theories where the global quantities associated with geometric objects are vectors or more complex mathematical entities. This is the case of fluid dynamics and continuum mechanics (where vector quantities such as displacements, velocities, and forces are associated with geometric objects). In this case, the deduction of the factorization diagram can be a difficult task, for one must first tackle a nontrivial classification task for quantities that have, in local form, a tensorial nature, and then disentangle the constitutive and topological factors of the corresponding equations. Moreover, for vector theories it is more difficult to pass silently over the fact that to compare or add quantities defined at different locations (even scalar quantities, in fact), we need actually a connection defined in the domain. To simplify things, one could be tempted to write the equations of fluid dynamics as a collection of scalar transport equations hiding within the source term everything that does not fit in an equation in the form of Equation (69), and apply to these equations the results of the analysis of the scalar transport equation. However, it is clear that this approach prevents the correct association of physical quantities with geometric objects, and is, therefore, far from the spirit advocated in this work. Moreover, the inclusion
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
45
of too many interaction terms within the source terms can spoil the significance of the analysis, for example, hiding essential nonlinearities. 9 Finally, it must be said that, given a field problem, one could consider the possibility to adopt a Lagrangian viewpoint in place of the Eulerian one that we have considered so far. The approach presented here applies, strictly speaking, only to a Eulerian approach. Nevertheless, the benefits derived from a proper association of physical quantities to oriented geometric objects extend also to a Lagrangian approach. Moreover, the case of moving meshes is included without difficulties in the space-time discretization described below, and in particular in the reference discretization strategy that will be introduced in the section on numerical methods.
III. REPRESENTATIONS
We have analyzed the structure of field problems aiming at their discretization. Our final goal is the actual derivation of a class of discretization strategies that comply with that structure. To this end, we must first ascertain what has to be modeled in discrete terms. A field problem includes the specification of a space-time domain and of the physical phenomena that are to be studied within it. The representation of the domain requires the development of a geometric model to which mathematical models of physical quantities and material properties must be linked, so that physical laws can be finally modeled as relations between these entities. Hence, our first task must be the development of a discrete mathematical model for the domain geometry. This will be subsequently used as a support for a discrete representation of fields, complying with the principles derived from the analysis of the mathematical structure of physical theories. The discrete representation of topological laws, then, follows naturally and univocally. This is not the case for constitutive relations, for the discretization of which various options exist. In the next sections we will examine a number of discrete mathematical concepts that can be used in the various discretization steps.
A. Geometry The result of the discretization process is the reduction of the mathematical model of a problem having an infinite number of degrees of freedom into 9As quoted by Moore (1989), Schr6dinger, in a letter to Born, wrote: "'If everything were linear, nothing would influence nothing,' said Einstein once to me. That is actually so. The champions of linearity must allow zero-order terms, like the right side of the Poisson equation, AV= --4rcp. Einstein likes to call these zero-order terms 'asylum ignorantiae'."
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C L A U D I O MATTIUSSI
one with a finite number. This means that we must find a finite number of entities, which are related in a known way to the physical quantities of interest. If we focus our attention on the fields, and think in terms of the usual continuous representations in terms of scalar or vector functions, the first thing that comes to mind is the plain sampling of the field functions at a finite number of points--usually called nodes--within the domain. This produces a collection of nodal scalar or vector values, which eventually appear in the system of algebraic equations produced by the discretization. Our previous analysis reveals, however, that this nodal sampling of local field quantities is unsuitable for a discretization, which aims at preserving the mathematical structure of the field problem, since this requires the association of 91obal physical quantities with geometric objects that are not necessarily points. From this point of view, a sound discretization of geometry must provide all the kinds of oriented geometric objects that are actually required to support the global physical quantities appearing within the problem, or at least, those appearing in its final formulation as a set of algebraic equations. Let us see how this reflects on mesh properties.
1. Cell-complexes Our meshes must allow the housing of global physical quantities. Hence, their basic building blocks must be oriented geometric objects. Since we are going to make heavy use of concepts belonging to that branch of mathematics called algebraic topology, we will adopt the corresponding terminology. Algebraic topology is a branch of mathematics that studies the topological properties of spaces by associating them with suitable algebraic structures, the study of which gives information about the topological structure of the original space [Hocking and Young, 1988]. In the first stages of its development, this discipline considered mostly spaces topologically equivalent to polytopes (polygons, polyhedra, etc.). Many results of algebraic topology are obtained by considering the subdivisions in collections of simple subspaces, of the spaces under scrutiny. Understandably, then, many concepts used within the present work were formalized in that context. In the later developments of algebraic topology, much of the theory was extended from polytopes to arbitrary compact spaces. The concepts involved became necessarily more abstract, and the recourse to simple geometric constructions waned. Since all our domains are assumed to be topologically equivalent to polytopes, we need and will refer only to the ideas and methods of the first, more intuitive version of algebraic topology. With the new terminology, what we have so far called an oriented pdimensional geometric object, will be called oriented p-dimensional cell, or simply p-cell, since all cells will be assumed to be oriented, even if not
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
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explicitly stated. From the point of view of algebraic topology, a p-cell rv in a domain D can be defined simply as a set of points that is homeomorphic to a closed p-ball B v = {x~ 9tv: Ilxll ~< 1} of the Euclidean p-dimensional space [Franz, 1968; Hocking and Young, 1988; Whitney, 1957]. To model our domains as generic topological spaces, however, would be entirely too generic. We can assume, without loss of generality, that the domain D of our problem is an n-dimensional differentiable manifold of which our p-cells are p-dimensional regular subdomains a~ (Boothby, 1986). With these hypotheses a p-cell ~v is the same p-dimensional "blob" that we adopted above as a geometric object. The boundary c~v of a p-cell ~v is the subset of D, which is linked by the above homeomorphism to the boundary c~Bv = {xegtv: ]]xl] = 1} of Bp. A cell is internally (externally) oriented when we have selected as the positive orientation one of the two possible internal (external) orientations for it. According to our established convention, we will add a tilde to distinguish externally oriented cells ~v from internally oriented cells zp. To simplify the notation, in presenting new concepts we will usually refer to internally oriented cells. The results apply obviously to externally oriented objects as well. In assembling the cells to form meshes, we must follow certain rules. These rules are dictated primarily by the necessity to relate in a certain way the physical quantities that are associated with the cells to those that are associated with their boundaries. Think, for example, of two adjacent 3-cells in a heat transfer problem; these cells can exchange heat through their common boundary, and we want to be able to associate this heat to a 2cell belonging to the mesh. To achieve this goal, the cells of the mesh must be properly joined (Figure 21). In addition to this, since the heat balance equation for each 3-cell implies the heat associated with the boundary of the cell, this boundary must be paved with a finite number of 2-cells of the mesh. Finally, to avoid the association of a given global quantity to multiple cells, it is desirable that two distinct cells do not overlap. A structure that complies with these requirements is an n-dimensional finite cell complex K. This is a finite set of cells with the following two properties: 9 the boundary of each p-cell of K is the union of lower dimensional cells of K. These cells are called the proper q-dimensional faces of ~v, with q ranging from 0 to p - 1. It is useful to consider a cell an improper face of itself. 9 the intersection of any two cells of K is either empty or is a (proper or improper) face of both cells. 1~ actual numerical problems p-cells are usually nothing more than bounded, convex, oriented polyhedrons in Ot".
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FIGURE 21. Proper and improper joining of cells.
This last requirement specifies the property of two cells being "properly joined." We can, therefore, say that a finite cell-complex K is a finite collection of properly joined cells with the property that if Zp is a cell of K, then every face of zp belongs to K. Note that the term face without specification of the dimension usually refers only to the (p - 1)-dimensional faces. We say that a cell complex K decomposes or is a subdivision of a domain D (written IKI = D), if D is equal to the union of the cells in K. The collection of the p-cells and of all cells of dimension lower than p of a cell-complex is called its p-skeleton. We will assume that our domains are always decomposable into finite cell-complexes and assume that all our cell-complexes are finite, even if not explicitly stated. The requirement that the meshes be cell-complexes may seem quite severe, for it implies proper joining of cells and covering of the entire domain without gaps or overlapping. A bit of reflection reveals, however, that this includes all structured and most nonstructured meshes, excluding only a minority of cases such as composite and nonconformal meshes. Nonetheless, this requirement will be relaxed later on or, better, the concept of a cell will be generalized, so as
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49
to include structures that can be considered as derived from a cell-complex by means of a limit process. This is the case of the finite element methods and of some of its generalizations, for example, meshless methods. For the time being, however, we will base the next steps of our quest for a discrete representation of geometry and fields on the hypothesis that the meshes are cell-complexes. Note that for time dependent problems we assume that the cell-complexes subdivide the whole space-time domain of the problem.
2. Primary and Secondary Mesh The requirement of housing the global physical quantities of a problem implies that both objects with internal and external orientation must be available. Hence, two logically distinct meshes must be defined, one with internal and the other with external orientation. Let us denote them with the symbols K and K, respectively. Note that this requirement does not imply necessarily that two staggered meshes must always be used, for the two can well share the same nonoriented geometric structure. There usually are, however, good reasons to also differentiate the two meshes geometrically. In particular, the adoption of two dual cell-complexes as meshes endows the resulting discrete mathematical model with a number of useful properties. In an n-dimensional domain, the geometric duality means that i of/~, and vice to each p-cell z ri of K there corresponds a (n - p)-cell ~,_p versa. Note that in this case we are purposely using the same index to denote the two cells, for this is not only natural but facilitates a number of proofs concerning the relation between quantities associated with the two dual complexes. We will denote with nv the number of p-cells of K and with tip the number of p-cells of K. If the two n-dimensional cell-complexes are duals, we have nv = fi,-v. The names primal and dual meshes are often adopted for dual meshes. To allow for the case of nondual meshes, we will call primary mesh the internally oriented one and secondary mesh the externally oriented one. Note that the discussion above applies to the discretization of domains of any geometric dimension. Figure 22 shows an example of the two-dimensional case and dual grids, whereas Figure 33 represents the same situation for the three-dimensional case.
3. Incidence Numbers Given a cell-complex K, we wish to give it an algebraic representation. Obviously, the mere list of cells of K is not enough, for it lacks all information concerning the structure of the complex, that is, it does not tell us how the cells are assembled to form the complex. Since in a cell-complex two cells can meet at most on common faces, we can represent the complex connectivity by means of a structure that collects the data about cell-face
CLAUDIO MATTIUSSI
50
FIGURE 22. The primary and secondary meshes, for the case of a two-dimensional domain and dual meshes. Note that dual geometric objects share a common index and the symbol that assigns the orientation. All the geometric objects of both meshes must be considered as oriented.
relations. We m u s t also include information concerning the relative orientation of cells. This can be d o n e as follows. Each oriented geometric object induces an orientation on its b o u n d a r y (Figure 4 and Figure 6), therefore, each (oriented) p-cell induces an orientation on its (p - 1)-faces. We can c o m p a r e this induced orientation with the default o r i e n t a t i o n of the faces as (p - 1)-cells in K. Given the i-th p-cell ~zP and the j-th (p - 1)-cell z~_ 1 of a complex K, we define an incidence number l-z~, ~ - 1 ] as follows (Figure 23)" 0 if z~_ 1 is not a face of~p _
=
+ 1 if ~ _ 1 is a face o f ~ and has the induced orientation (87)
-1
as above, but with opposite orientation
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
51
FIaURE 23. Incidence numbers describe the cell-face relations within a cell-complex. All the other 3-cells of the complex have 0 as their incidence number corresponding to the 2-cell ~.
This definition associates with an n-dimensional cell-complex K a collection of n i n c i d e n c e m a t r i c e s
Dp,p_~ = ([zp, z~_ 1-1)
(88)
where the index i runs over all the p-cells of K, and j runs over all the (p - 1)-cells. We will denote by D p , p _ ~ the incidence matrices of/s In the particular case of dual cell-complexes K and K, if the same index is assigned to pairs of dual cells, the following relations hold: f)v,p-1 = Dr- p + 1,.- p
(89)
It can be proved with simple algebraic manipulations [Hocking and Young, 1988], that for an arbitrary p-cell ~p, the following relationship holds among incidence numbers i
2 Z [-'Cp, "Cp_ 1-] ['Cp- 1, "C~_2-] --- 0 i j
(90)
Even if at first sight it does not convey any geometric ideas, from this relation there follow many fundamental properties of the discrete operators that we shall introduce below. The set of oriented cells in K and the set of incidence matrices constitute an algebraic representation of the structure of the cell-complex. Browsing through the incidence matrices, we can know everything concerning the
52
CLAUDIO MATTIUSSI
FICURE 24. Two adjacent cells have compatible orientation if they induce on the common face opposite orientations. The concept of induced orientation can be used to propagate the orientation of a p-cell to neighboring p-cells.
orientation and connectivity of cells within the complex. In particular, we can know if two adjacent cells induce on the common face opposite orientations, in which case they are said to have compatible or coherent orientation. This is an important concept, for it expresses algebraically the intuitive idea of two adjacent p-cells having the same orientation (Figures 23 and 24). Conversely, given an oriented p-cell, we can use this definition to propagate its orientation to neighboring p-cells (on orientable n-dimensional domains it is always possible to propagate the orientation of an n-cell to all the n-cells of the complex (Schutz, 1980)).
4. Chains Now that we know how to represent algebraically the cell complex, which discretizes the domain, we want to construct a machinery to represent generic parts of it. This means that we want to represent an assembly of cells, each with a given orientation and weight of our choice. A first requirement for this task is the ability to represent cells with the default orientation and cells with the opposite one. This is most naturally achieved by denoting a cell with its default orientation with rp and one with the opposite orientation with - r p . We can then represent a generic p-dimen-
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
53
sional d o m a i n Cp c o m p o s e d by p-cells of the complex K as a formal s u m np
Cp = ~
i wir p, r ip e K
(91)
i=1
where the coefficient w i can take the value 0, + 1, or - 1 , to d e n o t e a cell of the complex n o t included in Cp, or included in it with the default o r i e n t a t i o n or its opposite, respectively. This formalism, therefore, allows the algebraic r e p r e s e n t a t i o n of discrete s u b d o m a i n s as "sums" of cells. W e n o w m a k e a generalization, allowing the coefficients of the formal sum [ E q u a t i o n (91)] to take arbitrary real values wi~9t. To preserve the r e p r e s e n t a t i o n of the o r i e n t a t i o n inversion as a sign inversion, we a s s u m e that the following p r o p e r t y holds true w ~ ( - r ~i ) = - w ~
i
(92)
W i t h this extension, we can represent oriented p - d i m e n s i o n a l d o m a i n s where each cell is weighted differently (Figure 25). This entity is a n a l o g o u s ,
FIGURE 25. Given an oriented cell-complex (top), a p-chain (bottom) represents a weighted sum of oriented p-cells. Here the weights are represented as shades of gray. Note that negative weights make the corresponding cell appear in the chain with its orientation reversed with respect to the default orientation of the cell in the cell-complex.
54
CLAUDIO MATTIUSSI
in a discrete setting, to a subdomain with a weight function defined on it, thus it will be useful to give a geometric interpretation to the discretization strategies of numerical methods, such as finite elements, which make use of weight functions. In algebraic topology, given a cell complex K, a formal sum like Equation (91), with real weights satisfying Equation (92), is called a p-dimensional chain with real coefficients, or simply a p-chain Cp. If it is necessary to specify explicitly the coefficient space for the weights w~ and the cell-complex on which a particular chain is built, we write %(K, 9t). We can define in an obvious way an operation of addition of chains defined on the same complex, and one of multiplication of a chain by a real number, as follows:
% + %, = Z
t i
i
+E
i
i
~Cp ~ ~ 2 i
= E (w, +
i
(93)
i
WiT'Pi l_ E ('~'Wi)'~Pi i
(94)
With these definitions the set of p-chains with real coefficients on a complex K becomes a vector space Cp(K, ~) over 9t, often written simply as Cp(K) or Cp. The dimension of this space is the number np of p-cells in K. Note that each p-cell Zp can be considered an elementary p-chain 1.Zp. These elementary p-chains constitute a natural basis in Cp, which permits the representation of a chain by the np-tuple of its weights. Cp --- (W1, W 2 , . . . ,
Wnp )
(95)
Working with the natural basis, we can easily define linear operators on chains as linear extensions of their action on cells. In particular, this is the case for the definition of the boundary of a chain.
5. The Boundary of a Chain The boundary C~Zp of a cell Zp is by definition the collection of its faces, endowed with the induced orientation (Figure 4 and Figure 6). Remembering the definition of the incidence numbers, we can write t~'lSp =
np- 1 E ['l~p, "/~J_ 1 ] ' E J _ I j=l
(96)
where the index j runs on all the ( p - 1)-cells of the complex. Note that Equation (96) gives to a geometric operation, an algebraic representation based uniquely on incidence matrices. Since the p-cells constitute a natural basis for the space of p-chains, we can extend linearly the definition of c3 to an o p e r a t o r - - t h e boundary operator--acting on arbitrary p-chains, as
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
55
follows:
Thus the boundary of a p-chain is a (p - 1)-chain, and ~ is a linear mapping c~:Cv(K) ~ C v- I(K) of the space of p-chains into that of (p - 1)-chains. It can be proved [Hocking and Young, 1988], using Equation (90), that for any chain e v the following identity holds true: 0(c~cv) = 0
(98)
that is, the boundary of a chain has no boundary, a result that, when applied to elementary chains, satisfies our geometric intuition. The boundary of a cell defined by Equation (96) coincides practically with the usual geometric idea of the boundary of a domain, complemented by the fact that the faces are endowed with the induced orientation. The calculation of the boundary of a chain defined by Equation (97) can instead give a nonobvious result. Let us consider p-chains built with a set of cells that forms a connected p-dimensional domain (Figure 26). For some chains of this kind, it may happen that the result of the application of the boundary operator includes (p - 1)-cells, which we typically consider internal to the domain. In fact, it turns out that this represents the rule, not the exception, since an "internal" (p - 1)-cell does indeed appear in Equation (97), unless the sum of the weights received by it from the p-cells of which it is a face (the so-called cofaces of the (p - 1)-cell) vanishes. Obviously, this vanishing is true only for particular sets of weights, that is, for particular chains. Later, we shall build a correspondence between chains and weighted domains. In that context, the boundary of a weighted domain will be defined, and the result will turn out to be confined to the traditional boundary only for particular weight functions.
B. Fields A consequence of our traditional mathematical education is that when we hear the word field we tend to think immediately of its representation in terms of some kind of field function, that is, of some continuous representation. If we refrain from this premature association, we can easily recognize that the transition from what is observed to this kind of representation requires a nontrivial abstraction. In practice, we can measure only global quantities, that is, quantities related to macroscopic p-dimensional spacetime subdomains of a given domain. It is, however, natural to imagine that we could potentially perform an infinite number of measurements for all the
56
C L A U D I O MATTIUSSI
FIGURE 26. Given a p-chain cp (top) its boundary t~cr is a ( p - 1)-chain (bottom) that usually includes internal "vestiges" with respect to what we used to consider the boundary of the domain spanned by the p-cells appearing in the p-chain. Here the weights of 2-cells are represented as shades of gray and those of 1-cells by the thickness of lines.
possible subdomains. We then conceive this collection of potential measurements as a unique entity, which we call "the field," and we represent mathematically this entity in a way that permits the modeling of these measurements, for example, as a field function that can be integrated on arbitrary p-dimensional subdomains. Consider now a domain where we have built a mesh, say a cell-complex K. By so doing, we have selected a particular collection of subdomains, the cells of the complex K. Consequently we must (and can) deal only with the global quantities associated with those subdomains. The fields will manifest themselves on this mesh as collections of global quantities associated with these cells only. Of course, this association will be sensitive to the orientation and linear on cell assembly. This, in essence, is the idea behind the representation of field on discretized domains in terms of cochains.
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
57
1. Cochains
In algebraic topology, given an oriented cell-complex K and an (algebraic) field ~ , a function e p, which assigns to each cell Zpi of K (thought of as an elementary chain) an element c i of ~-, written ('r,p, r
= ci
(99)
and is linear on the operation of cell assembly represented by chains, that is, satisfies (Cp, c p) -
WiZp, c p = ~ wi(Vp, c p)
(100)
is called a p-dimensional cochain, or simply p-cochain c p. It can be written as cP(K, ~ ) or cP(K) to designate explicitly the complex and the algebraic field involved in the definition (when the complex is externally oriented, we will write cP(K) if the complex is explicitly mentioned, and ~.P if it is not). We will call ordinary cochains those defined on an internally oriented cell-complex, and twisted those defined on an externally oriented one [Burke, 1985; Teixeira and Chew, 1999]. We can readily see that this definition contains the essence of what we said above concerning the action of physical fields on domains partitioned into cell-complexes. The cochain, like a field, associates a value with each cell and the association is additive on cell assembly. Note that from Equation (100) there follows that ( - Z p , c p) - -(l:p, c p)
(101)
that is, as expected, the value assumed by a cochain on a cell changes sign with the inversion of the orientation of the cell. Thus, the only thing that must be added to the mathematical definition of a cochain to make it suitable for the representation of fields is the attribution of a physical dimension to the values associated with cells. With this further attribution the values can be interpreted as global physical quantities ( w h i c h - - w e stress once a g a i n - - n e e d not be scalars) and the corresponding entity can be called a physical p-cochain. All cochains considered in this work must be considered physical cochains, even if the qualifier "physical" is omitted. From Equation (100) we see that a cochain c p is actually a linear mapping Cp" Cp(K)---+ ~" of the space of chains Cp(K) into the algebraic field ~-, which assigns to each chain Cp a value (Cp, c p)
(102)
This representation emphasizes the paritetic role of the chain and of the cochain in the pairing. To assist our intuition, we can think of Equation
58
CLAUDIO MATTIUSSI
(102) as a discrete counterpart of the integral of a field function on a weighted domain, and this can suggest the following alternative representation for the pairing [Bamberg and Sternberg, 1988]
fc. cp
(103)
We can define the sum of two cochains and the product of a cochain by an element of ~-, as follows:
(c,, c" + c") = (c,, c') + (c,, c") (,~c,, c p) = 2(c,, c p)
(104)
(lo5)
This definition transforms the set of cochains in a vector space CP(K, ~ ) over f f usually written simply as CP(K) or Cp. A natural basis for this vector space is constituted by the elementary p-cochains, which assign the unity of ~- to a p-cell and the null element of f f to all other p-cells of the complex. The dimension of CP(K) is, therefore, the number np of p-cells in K, and on the natural basis we can represent uniquely a cochain as the np-tuple of its values on cells c" = (c ~, c~, . . . c " " ) L
c' = ( G ,
c") e g
(106)
With this representation, and with the corresponding one for a chain [Equation (95)], the pairing of a chain and a cochain is given by np (Cp, Cp) -- E i=1
Wi Ci
(107)
In the case of a physical cochain, the natural representation would be an np-tuple of global physical quantities associated with p-cells. For example, in a heat transfer problem the heat content 3-cochain 03 is represented by the fi3-tuple of the heat contents of the 3-cells ~3 I~2 = (Q J, Q2,..., Q~3)T
(108)
Q~ = Qc(~) = ('g~, 03)
(109)
where
The heat Q,. associated with a chain c-3 = Z~3 i=1 wi~ corresponds, therefore, to ~3
Qc = (~3, 07) = ~ wiQ~ i=1
(110)
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
59
FIGURE 27. The Tonti classification diagram of global electromagnetic physical quantities in terms of cochains. Note the presence of two null-cochains, corresponding to the absence of magnetic charge and to the absence of electric charge production.
Note the similarity with a weighted integral
Qc = ;~ wq~
(111)
The classification diagrams of physical quantities can be redrawn for a discretized domain, substituting the field functions with the corresponding cochains. For example, in electromagnetism we have the 1-cochain U 1 of electromagnetic potential, the 2-cochains (I)2 and ~2 of magnetic flux and electric flux, respectively, and the 3-cochain 0 3 of electric charge (to which we must add the zero 3-cochain 03 of magnetic charge and the zero 4-cochain ~4 of electric charge creation). The corresponding classification diagram is depicted in Figure 27.
60
C L A U D I O MATTIUSSI
Remark: It is sometimes argued that on finite complexes, cochains and chains coincide, since both associate numbers with a finite number of cells [Hocking and Young, 1988]. The two concepts are actually quite different. Chains can be indeed seen as functions that associate numbers with cells. The only requirement is that the number changes sign if the orientation of the cell is inverted. Note that no mention is made of values associated with collections of cells, nor could it be made, for this concept is still undefined. Before the introduction of the concept of chain we have at our disposal only the bare structure of the c o m p l e x - - t h e set of cells in the complex and their connectivity as described by the incidence matrices. It is the very definition of chain that provides the concept of an assembly of cells. Only at this point can the cochains be defined, which not only associate numbers with single cells--as chains d o - - b u t also with assemblies of cells. This association is required to be not only orientation-dependent, but also linear with respect to the assembly of cells represented by chains. This extension from weights associated with single cells to quantities associated with assemblies of cells is not trivial, and makes cochains a very different entity from chains, even on finite cell-complexes.
2. Limit Systems
The idea of the field as a collection of its manifestations in terms of cochains on the cell-complexes, which subdivide the domain of a problem, finds a representation in certain mathematical structures called limit systems. The basic idea is that we can consider in a domain D the set X of all the cell-complexes that can be built on it (with the kind of orientation that suits the field at hand). We can then form a collection of all the corresponding physical p-cochains on the complexes in Y . This collection can be considered intuitively the collection of all the possible measurements for all possible field configurations on D. Now we want to partition this collection of cochains in sets, with each set including only measurements that derive from a given field configuration. We define for this task a selection criteria based on the additivity of global quantities. At this point we can consider each of these sets a new entity, which in our interpretation is the field thought of as a collection of its manifestations in terms of cochains. We can define operations between fields, and operators acting on them, deriving naturally from the corresponding ones defined for cochains. For example, we can define addition of field and the analogous of traditional differential operators (gradient, curl, and divergence) in intuitive discrete terms. This allows an easy transition from the discrete, observable properties, to the corresponding continuous abstractions.
N U M E R I C A L M E T H O D S F O R PHYSICAL F I E L D P R O B L E M S
61
The reader is warned that the rest of this section is quite abstract, as compared to the prevailing style of the present work. The details, however, can be skipped at first reading, since only the main ideas are required in the sequel. The point is not to give a sterile formalization to the ideas presented so far, but to provide conceptual tools for the representation of the link existing between discrete and continuous models. Now, the mathematics. Consider the set ~ = {K~} of all cell-complexes that subdivide a domain D. Here the complexes are internally oriented, but they could be externally oriented ones as well. We will say that a complex Ke is a refinement of K~ - - written K~ < K p - - i f each cell of K~ is a union of cells of K~. The set Y is partially ordered by the relation <, and for any pair of complexes K~, K~ there exists a complex K~ in Y , which is a refinement of both [Whitney, 1957] (remember that our domains are homeomorphic to polytopes). This property makes of ~(( a directed set [Hocking and Young, 1988; Eilenberg and Steenrod, 1952]. For each complex K~ in X let us consider the set CP(K~) of all the physical p-cochains on K~, with a given physical dimension (with our choice of the space where the cochains take their values, CP(K~) is a vector space, but to keep things simple, we will consider only the group operation in the sequel). Whenever K~ < K~ in #f, there is a transformation fK~Kp:CP(K~) CP(K~) of CP(K~) into CP(K~) which, to each cochain eP(Kp), associates the cochain eP(K~) taking on each cell ~p of K~ the sum (with proper signs, to take care of orientation) of the values taken by eP(K~) on the cells of K~, which compose ~p. Physically speaking the transformations fKo/r are based on the additivity of the global physical quantities upon cell assembly. These transformations satisfy the following properties:
9f~K~ is the identity transformation for each K~ in Y, 9fK~K~fK~K~= f ~ ,
whenever K~ < K~ < K~.
If F denotes the collection {fK~K~} of all such transformations, and ~" the collection {CP(K~), K ~ e S } of all the physical p-cochains homogeneous relatively to the physical nature of the field, on cell-complexes in ~f, the pair {cgr, F} is called an inverse limit system over the directed set Y . Each set CP(K~) in the collection cgr is a group, and each f ~ , ~ in F is a homomorphism. The inverse limit 9roup Cr(K o0) of the system {cgr, F} is that subgroup of the direct sum ZxCP(K~) consisting of all sets {eP(K~)}, one element from each group CP(K~) for which fK, K,(cP(Kp)) = c"(K~) whenever K~ < Kp in #{. The group operation in CP(K o0) is defined naturally by the formula +
=
(112)
62
CLAUDIO MATTIUSSI
where the sum on the right indicates the group operation in each CP(K~). For each K~ in X there is a natural projection ~z~'CP(Ko~)~ CP(Kt3), defined by ~z~:~({eP(K,)}) = eP(Ktj). Each projection rcr, is a homomorphism. As anticipated we can interpret physically all of this, as follows. Each element {eP(K~)} of the inverse limit group CP(Koo) represents a physical field defined in the domain D, which associates physical quantities to p-dimensional geometric objects (let us call it a physical p-field, or simply p-field). The set {eP(K~)} is the collection of its manifestations (in terms of cochains eP(K~)) on the cell-complexes K ~ Y , which decompose D. The cochains that correspond to a given field can be recognized for they are linked by the functions in F. The group operation in CP(K o0) is the addition of fields. The projection rCK~associates to each cell-complex K~ and to each field {eP(K~)} the corresponding cochain eP(Kt3). In the example above we would have on a complex K, the heat content cochains Q~(K,) and each set {Qc(K~)} of the inverse limit group Q~(K oo) would be particular heat content field. To complete this exposition of limit systems, we will now anticipate some ideas related to the action of an operator between cochain spaces, that will be introduced in the next section. For each cell complex K, we will define an operator 6 ~ - - t h e coboundary o p e r a t o r - - w h i c h will be used to write topological laws in discrete form. This is a homomorphism of CP(K,) into C p+ ~(K~). Given the two inverse limit systems {c~P,F} and {~P+ ~,F'} over the directed set ~ (i.e., the inverse limit system constructed with the p-cochains and the (p + 1)-cochains on the cell-complexes that decompose D), we define a transformation 6 of {~P, F} into {cs ~, F'} based on the action of 6~. The transformation ,5 consists, for each K, in ~(F, of the transformation 6~, with the condition that whenever K~ < K~ in Y , the commutative relation ,SKJ,,,u,~ = fk, r~fK~ holds true, such that the sum on p-cells goes into the sum on (p + 1)-cells. Such a transformation ~ of (~P, F} into {~P+~,F'} induces a homomorphism 6a of the inverse limits groups CP(K~) into C p+ I(K~) as follows. If {cP(K~)} is an element of {~P, F), and K~ in Y is given, set c p+ ~ ( K , ) = 6K,(cP(K,)). Note that if K~ < Ka the above commutative relation tells us that fk, K~(cr+ ~(Ka)) = c p+ ~(K~). Thus {cP+~(K~)) is an element of Cp+I(K~). We define 6~({cP(K~)})= {cp§ I(K~)}. Since each element {qgP(K,)} of the inverse limit group CP(Koo) represents a p-field defined in the domain D, 6~ represents an operator, which transforms a p-field in a (p + 1)-field on D. We will see that it can be considered a way to define "differential" operators without the use of derivatives. All this shows how the idea of field can be considered a limit concept abstracted from a collection of discrete manifestations of the field on cell-complexes or, if you prefer, from a collection of potential measurements
N U M E R I C A L M E T H O D S FOR PHYSICAL F I E L D PROBLEMS
63
of global physical quantities. Correspondingly, a physical law concerning a field can be considered a collection of relations between the cochains that constitute the field. Note that the idea of field is an abstraction that remains at a higher logical level than actual measurements. So, we must take care not to treat a single cochain on a particular cell-complex as if it were a field (which is instead a class of cochains), for this would be an error of logical typing (like eating the menu card instead of the dinner) [Bateson, 1972].
C. Topological Laws Equipped with the concept of the cochain, we are now in a position to give topological laws a discrete representation. We have derived above the topological laws of electromagnetism in discrete form [Equations (56) through (62)]. The fundamental property of all these relations--shared by all topological laws--is that they equate a global physical quantity associated with a geometric object to another global quantity associated with its boundary. This appears even more clearly in the space-time formulation of the same laws (Equations (63) through (66), collected below for easy reference) ~b(8~) = O(V)
(113)
Q(c~ ) = 0(~ )
(114)
~'(c~S) = ~b(S)
(115)
~ ( c ~ ) = Q(~/)
(116)
If the domain is meshed with the primary and secondary cell-complexes we will have to substitute to the generic geometric objects appearing in Equations (113) through (116) the cells of K and K. Equations (113) through (116), then, become ~b(c~3) = 0(z3) Vz3 ~K
(117)
Q (c?'g4) = 0('~4) V~4 ~ K
(118)
~'(0z2) = ~b(~2)Vzz~K
(119)
~(c~3) = Q('g3) V'~36 K
(120)
Equations (117) through (120) have simple interpretations. For example, Equation (120) says that the charge associated with each 3-cell of the secondary mesh K equals the electric flux associated with the boundary of the 3-cell.
64
CLAUDIO MATTIUSSI
1. The Coboundary Operator Each of Equations (117) through (120) is a list of equivalences between global quantities. We can ask if the discrete representation of fields in terms of cochains can be used to write this list in a more compact way. A first step in this direction consists in writing each global quantity as a chain-cochain pairing. For example Equation (120) becomes (a~3 ' kI_/2) __ (~3, ~ 3 ) V~.3 (ER
(121)
However, this is still a list of equivalences of global quantities, not an equivalence of two cochains. On the other hand, we cannot equate directly the two cochains k]/2 and 0 3 because a domain and its boundary have different geometric dimensions, and the corresponding cochains belong consequently to two different spaces, in this case to C2(K) and C3(K), respectively. We can circumvent this problem by defining an operator 6, which, on a given cell-complex K, transforms a p-cochain c p into a (p + 1)-cochain 6e p, which satisfies the following relation
("Cp+ 1, I~cP) def__(~,~p+ 1, cP) V'Cp+ 1 ~ K
(122)
We can extend linearly this definition to arbitrary chains, as follows
(Cp+ 1, ~cP) def=(~Cp+ 1, cP)
(123)
In this way we have defined an operator 6, which is a linear mapping of the cochain space CP(K) into Cr+I(K) (to see how this operator acts on combinations of cochains, simply substitute e r + e r' or 2e r to e p on the left side and observe the effects on the right side). This operator is called the coboundary operator and is just the operator needed to construct the linear transformation 6~ of the inverse limit system CP(Ko~) into C r+ l(Ko~) (i.e., the transformation of p-fields into (p + 1)-fields) anticipated in the final paragraphs of the section devoted to limit systems. Let us apply the definition of this new operator to Equation (121). Particularizing Equation (122) for the electric flux cochain W2, we have
(124) which, substituted in Equation (121) gives
('~3, ~1.I_/2) .._._('~3, ~3) V~.3 ~/~
(125)
Since Equation (125) asserts the identity of the components of the two cochains in the natural basis representation, it affirms, in fact, the identity
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
65
of the two cochains. We can, therefore, thanks to the definition of the operator ~, write the topological law [Equation (116)] in terms of cochains only, as follows 6~ 2 = 0 3
(126)
The definition [Equation (123)] of the coboundary operator might seem quite abstract. However, it has a very intuitive meaning that can be exemplified as follows [Tonti, 1975]. Equation (124) can be rewritten substituting to &3 its expression in terms of incidence numbers. Exploiting the linearity of the chain-cochain pairing, after some reordering of terms, gives ('~3, (~@2) ._. 2 ['~3, "~] ('~, ~ 2 ) i
(127)
More generally, Equation (122) becomes (rp+l, 6e p) = ~ [rp+l, r/p] (~ip, e p) i
(128)
This means that the coboundary operator operates on a p-cochain e p and builds a (p + 1)-cochain 6e p, which assumes on each (p + 1)-cell rp+ 1 the global physical quantity associated by e p with the boundary of rp+ 1. This value is equal to the sum of the physical quantities associated by e p to the faces of rp+ 1 endowed with the induced orientation (Figure 28). In other words, the coboundary operator takes a quantity associated with the boundary of a geometric object and transfers it to the object itself. F r o m Equation (128) we see also that using the natural representation for the cochains, the coboundary operator admits the following matrix representation in terms of incidence matrices: 6c p = Dr+ 1,p" c r
(129)
2. Properties of the Coboundary Operator The coboundary operator enjoys a number of useful properties that are a discrete version of familiar properties of differential operators (in light of our discussion about limit systems, it is more appropriate to say that the properties of the differential operators follow from those of the coboundary operator). First, as a consequence of the relationship [Equation (90)] holding between incidence numbers (or, if you prefer, from the property c3(c~ev) = 0, holding for any chain, and the adjointness of boundary and coboundary operator), for any cochain e p we have 6(6ev) = 0 [Hocking and Young, 1988]. This property is reflected in the identities curl(grad (p)= 0 holding for any 0-field ~0, and div(curl A) = 0 holding for any 1-field A.
66
CLAUDIO MATTIUSSI
FtCURE 28. The coboundary of a p-chain is a (p + 1)-cochain, which assigns to each (p + 1)-cell the sum of the values that the p-cochain assigns to the p-cells which form the boundary of the (p + 1)-cell. Note that each quantity appears in the sum multiplied by the corresponding incidence number.
Next, on a pair of dual n-dimensional cell-complexes the following property of the coboundary operators acting at the same level and at opposite sides of the factorization diagram holds true. The coboundary transforming p-cochains e r in (p + 1)-chains of the primary complex is the adjoint, relative to a natural duality between cochains defined by suitable bilinear forms (.,.>, of the coboundary transforming (n-(p + 1))-cochains in (n - p)-cochains of the secondary complex [Mattiussi, 1997]. For example, in Figure 29, this is the case of the operators acting in 6U 1 and 61t~/2,which satisfy
<6u ~, q'~> =
(~30)
For generic cochains this property corresponds to <(~cp ~n-(p+ 1)> = <ep (~n-(p+ 1)>
and is expressed in terms of incidence matrices by Equation (89). The corresponding property for differential operators is the (formal) adjointness of - g r a d and div, and of curl and - c u r l . The nondegenerate bilinear forms,
N U M E R I C A L M E T H O D S FOR PHYSICAL F I E L D PROBLEMS
67
which put in duality the cochain spaces in Equation (130), are in the natural and ( ( I ) 2 ~ / 2 ) __ basis representation ( Ux, 0 3 ) "--" '~'~n!ln3 ~iQi ~n2 "-"/'/2qSjffj where we have assumed that dual cells share the same index). ~-1 These bilinear forms are discrete counterparts of the energy integrals for the corresponding local field quantities on which the adjointness of the differential operators is based. In summary, by adopting the coboundary operator for the discrete representation of topological laws, and a pair of dual cell-complexes as primary and secondary meshes, one automatically builds into the resulting numerical method a number of important properties of the continuous mathematical model. By so doing, contrary to what happens within many numerical methods, one is not forced to check after the discretization has been performed if these properties are satisfied, or to enforce explicitly these properties as additional constraints in the course discretization phase. Note that the prescription to write the topological equations in terms of the coboundary operator does not imply the use of some exotic mathematical entity. It means simply that you adopt the correct association of global physical quantities with oriented geometric objects, and that you write the topological equations in integral form, equating the global quantity associated with each cell with that associated with its boundary. From this point of view, when all the formal properties have been proved, to say that we are using the coboundary operator is simply a shortcut to signify that this sequence of steps is being executed.
3. Discrete Topological Equations Applying the definition of the coboundary operator 6, we can rewrite the topological laws of electromagnetism [Equations (113), (114), (115), and (116)] as relations between physical cochains. The steps are those leading from Equation (116) to Equation (126). The result is ~(I) 2 --- 0 3
(131)
6Q3 = ~4
(132)
6U 1 = ~2
(133)
6~2 = i)3
(134)
We can thus redraw the space-time classification diagram of Figure 14 in terms of cochains and coboundaries (Figure 29). Note the presence in the diagram of Figure 14 and in Equations (131) and (132) of the zero 3- and 4-cochains on K and K, respectively. Note also that contrary to the traditional Maxwell's equations in differential vector notation, where positive and negative terms appear, in the
68
CLAUDIO MATTIUSSI
FIGURE 29. The Tonti classification diagram of electromagnetic physical quantities in terms of cochains, showing the topological laws in terms of coboundary operator.
cochain-coboundary notation, all signs are positive. This happens because the coboundary operator automatically takes care of the signs, by considering values on the boundary as endowed with the induced orientation. The presence of negative signs in the traditional form of these equations is due to the fact that the default orientation of a geometric object in the mathematical definition of an operator, and in the traditional definition of a physical quantity, may not agree. For example, the term - g r a d V in Equation (11) defining the electromagnetic potential is due to the fact that in mathematics the default orientation of points is as "sinks," so that the boundary of an internally oriented line is the endpoint "minus" the starting point, whereas the traditional definition of the scalar potential V implies a default orientation of points as "sources" (inherited, in fact, from the default external orientation of volumes to which electric charge is associated).
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69
Remember, also, in considering the meaning of signs, that the quantities are associated to space-time objects, and not merely to geometric objects in space. D.
Constitutive Relations
We said above that the constitutive relations do not lend themselves to a natural discrete representation. Nonetheless, some kind of discrete constitutive link must be given in order to complete the discretization of the physical field problem, and to arrive finally at a finite system of algebraic equations that can be subjected to the action of an algebraic solver. The task of finding the discrete constitutive link constitutes, in fact, the central problem of the discretization step. We will consider later in detail a number of possible approaches to this task, mainly inspired from existing numerical methods. For the time being we shall limit ourselves to a generic analysis of the structure that this representation must possess. Given the discrete representation of fields in terms of cochains that was presented above, the discrete constitutive links must be operators linking cochain spaces (usually an ordinary cochain space to a twisted one, or vice versa), like F 1 9CP(K) ---. C q ( K )
(135)
F 2 "C"(K) ---. CS(K)
(136)
Usually, these operators are discrete links whose structure is directly inspired by that of the local ones between field functions. For example, if we denote with ~d the electric part of the electric flux 2-cochain, and with (De the electric part of the magnetic flux 2-cochain, the electric constitutive relation [Equation (14)] can be given an approximate discrete representation as an operator F~ as follows:
~d= Fe((De)
(137)
If the required link goes from (De to q,d, we shall represent it as
(De._ F_t(~/~d)
(138)
instead of (De F [ l(t~d), to allow for discrete operators obtained by direct discretization of the local link going from E to D, without limiting the choice to those obtained by inverting the link F~. Analogous representations hold, in both directions, for the magnetic constitutive relation [Equation (15)] and the generalized form of Ohm's laws [Equation (16)], the discrete _
.
70
CLAUDIO MATTIUSSI
version of which we will write as
F.(~ ~)
(139)
0_.j= F~(~ e)
(140)
~=
whereas the discrete links in the opposite direction will be written as ,st,',= v _ , ( ,~ b)
(141)
~e=r~-,(OJ )
(142)
These links are usually linear operators between cochain spaces, and can, therefore, be given a natural matrix representation that we will represent in the case of Equation (137) as follows qja=
F(i)e
(143)
with analogous renderings for the other electromagnetic constitutive links considered above. The widespread use of a linear discrete constitutive link stems from the desire to obtain a linear system of equations by composition of the discrete constitutive operator with the coboundary operator (which is a linear operator between cochain spaces). This choice, however, does not imply that the local constitutive relation from which they derive also be linear. 11 When the local constitutive relation is nonlinear, whereas the discrete constitutive equation is, this last link must be considered as obtained from the former by means of some kind of linearization technique within the numerical procedure. Note that links of the kind in Equations (135) and (136) have an even broader scope than may seem at first sight. Consider, for example, a time-dependent problem to be solved in a domain D during a time interval I. The space-time domain of the problem is the cartesian product D x I, which is decomposed by the primary and secondary cell-complexes K and K. Usually the decomposition is the product of a spatial decomposition K s of D by a decomposition K t of I (or K s and K t, respectively). With this hypothesis Equation (135), for_ example, includes relations_ linking the values taken_ by C q on all q-cells of K s for all time instants of K t and all (q - 1)-cells of K s for all time intervals of K,, to the value taken by C r on all p-cells of K s for all time instants of K t and all (p - 1)-cells of K s for all time intervals of K t. Of course, most of the time constitutive equations are very simple, particular cases of this general relation. A very common case, for example, with a dual cell-complexes and the natural dual indexing of cells, would be a diagonal operator. However, we will see later that more general cases can 11In fact, this will usually not be the case, since, being that topological equations are always linear, nonlinearities present in the field equations are due to the constitutive terms, where they can be found when the field equations have been factorized.
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71
FIGURE 30. The Tonti discrete factorization diagram of electromagnetism, with the fields represented in term of cochains, the topological laws in terms of coboundary operator, and the constitutive links as operators between cochain spaces.
be usefully considered. In particular, it might well happen that the actual discrete constitutive link is never explicitly determined, being obtained as a result of an algorithm, for example, an optimization procedure, taking as input a given known cochain, and giving as a result the cochain linked to the former by the constitutive relation. The availability of a discrete representation for the constitutive relations allows the redrawing of the complete factorization diagram in discrete terms. For the case of electromagnetism, the resulting factorization diagram is depicted in Figure 30. Note the distinction between the parts of the cochains referring to time instants and those referring to time intervals, and the corresponding one for the action of the coboundary operator. This distinc-
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CLAUDIO MATTIUSSI
tion is particularly useful in view of the application of the diagram to numerical methods, but remember that it applies only to space-time meshes obtained as cartesian products of separate discretizations of the space and time domains.
E. ContinuousRepresentations The last few sections showed how to represent discretized domains and subdomains, fields and topological laws in terms of cell-complexes, cochains and coboundary operator. This can be applied straightforward to obtain a formal treatment of the finite volume method, which writes balance and conservation equations over subdomains that are actually simple cells, such as /~) (curl E + ~-~-Bt)= 0
(144)
and f
div B - 0
(145)
3
which, after application of the integral theorems of vector calculus, become E +
-~- = 0
(146)
f,~ B = 0
(147)
1:2
2
1:3
The case of finite element methods, however, does not fit equally well in a representation built on these purely discrete concepts. For example, a weighted residual formulation of Equation (144) and (145) is w. c u r l E + - - ~
=0
(148)
f wdivB=O
(149)
3
3
where w is a vector valued weight function, w is a scalar one, and r 3 is a regular three-dimensional domain. After integration by parts, Equations (148) and (149) become curl w. E 3
wx E + r3
;grad 3
w.--~ = 0 3
.
0
(150)
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
73
TABLE 1 TABLE OF CORRESPONDENCES BETWEEN DISCRETE AND CONTINUOUS CONCEPTS
Discrete
Continuous
p-cell Zp boundary of a p-cell c~zv p-chain ep p-skeleton of a cell-complex K v p-cochain ep pairing p-chain-p-cochain (%, ep) coboundary operator 6 discrete generalized Stokes theorem
p-dimensional domain Dv boundary of a p-dimensional domain c~Dv p-weighted domain wv collection of p-weighted domains Wp p-form cop p-weighted integral of a p-form ~wcop exterior differential op. d continuous generalized Stokes theorem ~Dp+la~P= ~v +1coP continuous Green's formula (integration by parts) Sw~+~d~o~= S~.... ~o~ continuous topological equation
('Cp+ 1, r
-- ((~'Cp+ 1,c~)
discrete Green's formula (summation by parts) (%+ 1, ~cP) = (0%+1, c') discrete topological equation (C~Cp+1, aP) = (%+ 1, bY+1), Vcp+1, flap : bp+ x
which do not convey the immediate geometrical meaning of Equations (146) and (147). The weighted residual technique shown above in action, taken as representative of the strategy adopted by finite elements methods, permits nonetheless a geometric interpretation that parallels that of finite volume methods. To display this analogy, we will introduce some continuous concepts that correspond to the discrete ones introduced so far in the representation of geometry, fields, and topological laws (Table 1). Note that the aim of the present section is not the construction of an abstract correspondence between discrete and continuous concepts. The inspiration for the parallelism comes from (and is instrumental to the application to) numerical methods. The actual formal justification of a particular correspondence, that is, the proof that in the limit the discrete concept goes into the continuous one, may be very difficult, or even impossible. In this case we will simply be confident in the heuristic interpretative value of the correspondence thus built. To parallel the presentation of discrete representations made above, we should ideally deal first with continuous models for the domain geometry. It turns out, however, that a preliminary discussion of continuous field representations paralleling cochains is preferable. The only concept that we must suppose available is that of n-dimensional domain of integration D,
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MATTIUSSI
contained in the domain D, which can be considered as an n-dimensional differentiable manifold and usually is merely a regular subdomain of 9t". 1. Differential Forms Given expression such as Equations (150) and (151), if we want to find a geometric interpretation for the weighted residual methods, it is clear that we will have to deal with integrals. Strictly speaking, the "thing" that is subjected to integration on oriented p-dimensional domains is a p-dimensional differential form (usually simply called a p-form) [Deschamps, 1981; Warnick et al., 1997]. If the domain is internally oriented, we speak of an ordinary p-form, which we denote by cop, otherwise the p-form is called twisted and is denoted by (5p [Burke, 1985]. We said above that p-cochains associate values with p-cells and that this association is additive on the sum of cells. This is true also for the integration of p-forms on p-dimensional domains. In fact, a p-form on a cellulizable domain D gives rise to a p-cochain eP(K~) on each cell-complex K~, which subdivides D, since it associates with each cell Zpi a value c', where ci= ~
coP
(152)
J~ Note how this parallels the original definition of the action of a cochain, as associating with the cell the value
c'= (~/~,c~)
(153)
To emphasize this parallelism Bamberg and Sternberg (1988) call p-forms also incipient p-cochains. In fact, we can think of the p-form coP as a particular representation of an element {cP(K~)} of the inverse limit group CP( K o~). The particular cochain cP(K~) is then the projection of coP on the cell-complex K~. Given the correspondence of Equation (152) with Equation (153), we can ask what corresponds to the more general pairing of chains and cochains (Cp, cP). At first sight, we could consider an integral on a chain, using the definition = fc~Pfz
= wi z wp copdef2if
(154)
This is, in fact, the most general definition of integration on manifolds usually given in textbooks [Choquet-Bruhat and DeWitt-Morette, 1977; Flanders, 1989]. A bit of reflection, however, shows that this extension of the concept of integral, from p-dimensional domains to p-chains, does not fulfill our needs, since the integrals appearing in weighted residual express-
N U M E R I C A L M E T H O D S F O R PHYSICAL F I E L D P R O B L E M S
75
ions such as Equations (148) and (149), in the language of differential forms, correspond to f~ woo~
(155)
p
where w is a weight function defined on Zp. To find a way out of this impasse, we could think of Equation (155) as derived from the right-hand side of Equation (154) by a limit process that considers chains built on finer and finer meshes, or, alternatively, we could reconsider the way a Riemann integral is evaluated by partitioning the integration domain. We will pursue both lines of thought in the next subsection. One could at this point argue that finite element methods do not actually use formulations based on differential forms, since expressions such as Equations (148) and (149) have a scalar or vector field function in place of the differential form cop. This is, however, merely an unfortunate consequence of the historical development of vector calculus as applied to physics. The reduction of what are actually differential p-forms, to scalar and vectors only, in fact makes things more difficult to understand physically and represent geometrically. One has only to browse through books such as Burke [1985], Schouten [1989] and Misner et al. [1970], and papers such as Warnick et al. [1977], with their fascinating geometric representations of forms and integration, to convince ourselves of this fact.
2. Weighted Integrals Although the idea behind the operation is quite intuitive, to actually integrate a p-form on a differentiable manifold one must face some technical problems. For example, to define an analogy of Riemann integration, one must represent the integration domains and their partitions. This problem is usually circumvented thanks to the presence of a collection of m a p s - - t h e so-called coordinate m a p s - - o f a collection of open sets of the manifold where the integration takes place, onto open subsets of a suitable Euclidean space [Bishop and Goldberg, 1980]. This allows the definition of the pullback of a form from the manifold to the Euclidean space [Burke, 1985], where the familiar machinery of Riemann integration is usually invoked. To justify this step, it is often said that the differential form subject to integration in the manifold becomes, after pullback, an ordinary scalar function in Euclidean space, while the integration domain within the manifold, becomes an Euclidean domain, which can be easily partitioned into pieces of known extension. But how does it happen that a differential form within the manifold becomes an ordinary function in the Euclidean space? There is, in fact, a step that is usually passed over silently in forming
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the Riemann sums that define the integral. This step is the pairing of the pulled-back form with a collection of multivectors, which represent the parallelepipeds partitioning the domain in the Riemann integration procedure. This concept of multivector just introduced requires a brief digression. The calculus of differential forms is built on the algebra of forms, which defines forms as linear functions defined on spaces of multivectors. To this end one starts from a vector space and defines first the concept of p-vector vv [Birss, 1980]. You can think of a p-vector as defining a p-dimensional oriented domain within an affine space [-Tonti, 1975]. Thus, a 1-vector Vl is the familiar vector and defines an oriented segment along a line; a 2-vector v2, or bivector, defines an oriented surface on a plane; a 3-vector, v3 or trivector, defines an oriented volume; and so on, up to the maximum dimension allowed by the affine space. 12 Paralleling the distinction between internal and external orientation for geometric objects, we can define ordinary multivectors, corresponding to internally oriented geometric objects, and twisted multivectors, corresponding to externally oriented geometric objects [Burke, 1985] (Figure 31). Note that p-vectors constitute in turn a vector space, and that we can define the extension of a p-vector without recourse to metric concepts. Given the concept of multivector, one can define an algebraic p-form as a linear function on the space of p-vectors, with values in an algebraic field. From this definition it follows that the pairing of a p-vector vv and a p-form cop gives a value exactly like the pairing of a chain and a cochain (see Misner et al. [1970] and Warnick et al. [1997] for fascinating geometric illustrations of this pairing). This analogy suggests the following representation of the pairing (v p, coy)
(156)
Given these steps, to define the Riemann integral of a p-form in a Euclidean space, the integration domain D p is subdivided in a collection Vp of p-dimensional parallelepipeds, which can be thought of as p-vectors vvi [Dezin, 1995]. The corresponding Riemann sum is, therefore, S = ~ (v/p, cop)
(157)
i
In the sequence of domain partitions considered in the Riemann integration process, the maximum p-vector extension tends to zero, and the correspond12Actually, things can be more complex, due to the fact that, for example, in four-dimensional space a generic multivector can be compound [-Schouten, 1989; Tonti, 1975]. However, compound multivectors are not required to represent common geometric objects [Birss, 1980].
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
77
FIGURE 31. Ordinary and twisted multivectors correspond to internally and externally oriented geometric objects, respectively. Here the case of a three-dimensional vector space is represented.
ing limit, existing for suitable regularity conditions, is the integral of the form
~Dp (Dp
We see, therefore, that in building a Riemann sum we actually pair the differential form with a chain composed by the collection Vp of multivectors, which partition the domain. An obvious extension paralleling that which assigns real weights to cells in the formation of chains consists of using collections of weighted multivectors. This can be done assigning a scalar
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CLAUDIO
MATTIUSSI
weight function w on the integration domain. We can then define the new Riemann sums as follows
s = Z (w(~')v~, co~)
(158)
i
i of the domain decomposiwhere ~i is a point belonging to the p-vector Vp tion. Under the usual regularity conditions, the value of the limit of the sequence of Riemann sums with decreasing maximum extension of the i This limit multivectors, does not depend on the actual position of ~ in Vp. is the weighted integral, and can be denoted by
fw COp
(159)
p
This symbolism emphasizes the fact that the function w weights the integration domain, and is not an ordinary function that multiplies the form. This can be thought of as saying that integrals of the kind [Equation (159)] (which includes as particular cases those appearing in Equations (148), (149), (150), and (151)) must be considered Stieltjes integrals, and not ordinary integrals of the products of functions [Lebesgue, 1973]. If the integral is defined on a regular p-dimensional domain that lies in a manifold, we can, as anticipated, pull back the form and apply the procedure just described for the integration in Euclidean spaces. It is, however, instructive to consider the possibility of going the other way, that is, to push forward the multivectors that partition the Euclidean domain [Burke, 1985]. This can be thought of as providing a decomposition of the integration domain in the manifold. There are actually some technical difficulties, since the vectors thus pushed forward do not "belong" to the manifold but to its tangent space [Burke, 1985]. We can circumvent this problem, saving the heuristic value of the idea, thinking, for example, of manifolds that are subsets of a suitable Euclidean space, so that tangent multivectors are actually tangent parallelepipeds that approximate the true image of the Euclidean parallelepiped on the manifold (for example, see Figures 8.24 and 8.25 and related discussion in Bamberg and Sternberg [1988]). To this "decomposition" of the integration domain in the manifold apply all the considerations made above for Riemann sums and weighted integrals, and in particular the role of the function w in Equation (159) in weighting the "cells" of the decomposition of the domain considered in a Riemann sum, that is, in giving a collection of finer and finer chains, which can be thought of as the continuous counterpart of a chain.
3. Differential Operators To develop further the correspondence between differential forms and cochains, let us consider the action of the coboundary operator. The
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
79
definition [Equation (122)] of the coboundary operator on the cochains of a cell-complex K was given in order to allow the transition from a topological equation in the form
<(~72p+1, aP> -- <'Cp+ 1, b P +
1 > V'~p+ 1 ~
K
(160)
to the following direct relation between cochains gap __ bp+ 1
(161)
We can mimic the definition [Equation (122)] of the coboundary operator in the terminology of differential forms. We obtain
fv
p+1
dc~ deYf
Vp+1
c~
+1
cD
(
where d is an operator transforming p-forms into (p + 1)-forms. This operator is called the and inherits the property of the coboundary operator of allowing the transition from Equation (160) to Equation (161), by transforming a topological equation given in integral form, such as
exterior differential,
f
Vp+1
c~P=fo flP+IVDp+I~D p+1
(163)
into dc~p = tiP+ x
(164)
Note that usually the exterior differential is defined in terms of derivatives of the form's components, whereas Equation (162) constitutes an intrinsic definition [Isham, 1989] which, as emphasized by one of the creators of the calculus of forms, 13 does not require the existence of the derivatives of the form's components. The generic operator d defined by Equation (162), combines in a unique operator the action of the familiar differential operators gradient, curl, and divergence, which can also be given an intrinsic definition. Remembering that we call that which corresponds to a quantity associated with p-dimensional geometric objects, we can give the following definitions. The gradient operator acts on 0-fields and gives 1-fields, which satisfy
p-field
fD grad q0 def = f• 1 l a"On
q~VD 1 c D
(165)
D1
congoit donc la possiblit6 de d6finir la d6rivation ext6rieure comme une op6ration
autonome,ind@endante de la d6rivation classique." [Cartan, 1922].
CLAUDIO MATTIUSSI
80
the curl operator acts on 1-fields and produces 2-fields according to
f D CUrlA aef f oD2 A VD2 C
(166)
and the divergence operator acts on 2-fields and gives 3-fields satisfying
f
divBaejfBVD3~D 3
(167)
D3
It is worth noting that the property 66 = 0 of the coboundary operator is reflected in the property dd = 0 of the exterior differential operator, which in turn corresponds to curl grad = 0 and div curl = 0 in vector calculus notation. Given its properties, the exterior differential d, appears as the equivalent of the limit operator 6o0 defined at the end of the section on limit systems. No wonder then that its definition can be based on global concepts. Of course, given the additivity of global physical quantities, and the telescoping property following from the opposite orientation induced on the common boundary by adjacent, coherently-oriented domains, if the definition [Equation (162)] of d (and those Equations (165), (166), and (167) of the traditional differential operators of the vector calculus) is enforced in the small, it holds for every geometric object. This is why in textbook expositions, the definitions [Equations (165), (166), and (167)] are applied to infinitesimal one-, two- and three-dimensional rectangles, to derive the definition in local terms of the operators. This gives the familiar expressions in terms of derivatives, but our approach shows that these operators have a more general significance. The three-to-one relation of the differential operators of vector analysis with the exterior differential of forms, stems from the already signaled limitations of representation in terms of vectors alone, which hide the true "p-nature" of p-fields. Our treatment reveals, for example, that an expression of the kind curl (curl A)
(168)
is meaningless as such, for the 2-field produced by the first application of the operator, cannot be operated onto by the second operator. The actual expression should, therefore, be actually something like curl (k(curl A))
(169)
where the intermediate operator k represents an operator, for example, a constitutive link, which transforms a 2-field in a 1-field. (which, if k is a
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
81
constitutive operator, is usually endowed with a different kind of orientation with respect to A). Using differential forms and the exterior differential it is possible to rewrite in a compact way the equations of electromagnetism. One starts by grouping everything related to a given space-time geometric object in a unique differential form. Thus, remembering the classification of electromagnetic quantities defined above, one has an ordinary 2-form--the electromagnetic 2-form F 2, which "groups" E and B, or, better, the local counterparts of c e and ~bb--and a twisted 3-form--the charge-current 3-form ~3, grouping J and p. The local conservation of magnetic flux is expressed by dF 2 - 03, and that of electric charge by d j 3 = ~4. The charge-current potentials H and D go into a twisted 2-form (~2, related to j3 by d(~ 2 = ja, whereas the electromagnetic potentials go into an ordinary 1-form A 1 satisfying dA 1 - F 2. Since we are at it, the constitutive relations are expressed by a mapping between differential forms, for example, the electric and magnetic constitutive relations are expressed by (~2= z(F2), where Z is a generic operator from the space of ordinary 2-forms to that of twisted 2-forms (as detailed below, often erroneously identified with the Hodge star operator). The construction of the corresponding factorization diagram in terms of differential forms is straightforward.
4. Spread Cells Let us now go back to weighted integrals, and combine their properties with those of the newly defined differential operator. We have at last with ~wpcop an expression fully correspondent, in a continuous setting, to the chaincochain pairing (ep, cP). This is a bilinear pairing with respect to which the boundary and coboundary operators are mutually adjoint, satisfying the relation (Cp+~, 6c p) = (C3Cp+1, cP) 9In a similar way we can define the adjoint of the exterior differential as the boundary of the weighted domain. Formally this produces
fw
p+l
dcoP=
fo
cop
(170)
Wp+l
and can be given an explicit expression in terms of differential forms [Bamberg and Sternberg, 1988]. Since we are interested in its application within the weighted residual method, to interpret formulas such as Equations (148) and (149), let us see instead how this appears in the familiar language of vector calculus (remember that the exterior differential operator d corresponds to the gradient, curl, or divergence operators, depending on the kind of field under consideration). Integrating by parts the expression that corresponds to the left side of Equation (170) when d is the divergence
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C L A U D I O MATTIUSSI
operator, we have
wD-f~
f~ w d i v D = f ~ 3
~3
gradw. D
(171)
3
where the 3-cell z 3 can be taken as the support of the weight function w. 14 The right side of Equation (171) can be considered to correspond to that of Equation (170), that is, the expression for the "boundary" of a weighted three-dimensional geometric object. In other words, we can give the following formal definition [Mattiussi, 1997]
f~ Ddef;wD-f~
wz3
(wz3)
~3
gradw-D
(172)
3
where with we represent a weighted 3-cell. Note that, as anticipated, speaking of the boundary of chains, this "boundary" includes actually an integral on the whole 3-cell z 3, and not only on c?z3 but in the particular case of a weight function, which is constant on its support. We can, therefore, give the following geometric interpretation to the corresponding weighted residual formulas. The weight function w defines the continuous counterpart of a chain. We can think of it as a "spread" or "smeared out" cell [Mattiussi, 1997], to be compared with the "crisp" cells considered so far, which can be characterized by a weight function that is constant on its support [Ofiate and Idelsohn, 1992] (Figure 32). When an expression such as
f wdivD=fwp ~
(173)
is written within a finite element formulation of an electromagnetic problem, and the 1.h.s. is integrated by parts to get
i wD-f 1:3
3
gradw.D=f
3
wp
(174)
we can consider this last formula as the expression of the balance between the electric charge associated with the corresponding spread cell and the electric flux associated with the boundary of that spread cell, that is
;~,w~a)D=fw~3P
(175)
If the weight function is proportional to the characteristic function of a cell, that is, it is constant on the cell and is zero outside, the second term of ~4The support of a function is the closure of the set of points where it does not vanish [Bossavit, 1998a].
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83
FIGURE 32. Weight functions, which are constant on their support, define crisp cells (left). Generic weight functions define instead the continuous counterpart of a chain that can be thought of as a spread cell (right).
the 1.h.s. of Equation (174) vanishes, and the finite element method corresponds to the finite volume method [Fletcher, 1984; Ofiate and Idelsohn, 1992]. Otherwise, the finite collection W = {w~} of weight functions used within a weighted residual finite element formulation can be thought of as defining a continuous counterpart of a cell-complex, composed by spread cells. Of course, these spread cells usually overlap, whereas the p-cells of a cell-complex meet at most on lower dimensional cells. However, if the weight function constitutes a partition of unity in the domain [Belytschko et al., 1996], something of the spirit that dictated that request for cellcomplexes remains valid, since the sum of the physical quantities associated with the spread cells of W equals the amount of that quantity associated with the entire domain. Note that the role of integration by parts, or, if you prefer, of Green's formulas, is interpreted geometrically as defining implicitly the boundary of a spread cell. For this reason, the corresponding discrete formula [Equation (123)] can be called the discrete Green's formula or the summation by parts formula. It is worth emphasizing that this summation by parts formula, contrary to those used in the context of compact finite difference methods [Bodenmann, 1995; Lele, 1992], is based on topological concepts only, and does not require the preliminary definition of an inner product. Moreover, the summation by parts formula [Equation (123)] is automatically satisfied adopting a discretization based on cell-complexes,
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chains, cochains, and the corresponding operators, and, therefore, need not be imposed explicitly on the discrete operators that substitute the differential ones. The relation corresponding to Equation (171) for the case of twodimensional domains is
wE-f~
f, w c u r l E = ~ 2
~2
gradwxE
(176)
2
where E is a generic 2-field. This leads to the following formal definition
fo (r
E aef f ,~ wE -- f ~ g r a d w x E 2)
"C2
(177)
2
for the boundary of a spread 2-cell, to be used, for example, to enforce the following relation. E + ~-'~2)
= 0
(178)
~2 C3t
An expression such as Equation (177) would, however, find application within the finite element formulation of a three-dimensional problem, only if a peculiar kind of discretization were defined for the domain, mixing discrete and continuous concepts. An example of such a discretization would be a collection of weight functions defined on the 2-cells, of a cell-complex that subdivides the three-dimensional domain of the problem. For weighted one-dimensional domains the following definition would apply q0 = (wrl)
wq~ rl
grad wq~
(179)
1
As above, its use within a numerical method requires a mesh including a collection of spread 1-cells distributed within the domain of the problem. These kinds of formulations, mixing continuous and discrete concepts in the construction of the meshes are not presently used in the numerical practice. Instead in an n-dimensional domain, only n-dimensional weighted integrals are considered, such as the first term in the left side of Equation (148) in place of that of Equation (176). In this case, one can still think that the vector w(~), where ~ is a point within the support of the weight function w, defines locally a weight for bivectors orthogonal to w(~) (if the entities subjected to weighted integration are 2-fields) or vectors parallel to w(~) (if the entities are 1-fields). Thus some remnant of the geometric meaning of the weighted residual equation is still present in these formulations. Note that if the well-known integrability conditions hold, the support of w can be
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
85
thought of as sliced into a collection of spread cells. 15 For example, irrotational weight functions w(~) define a collection of surfaces orthogonal to the field w, whereas solenoidal weight functions define a collection of lines along it. Note that these cases correspond to the absence, in the expression for the boundary of the corresponding weighted domain, of terms that are integrated on the interior of the support of the weight function (this is the continuous counterpart of the presence of "interior" cells in the boundary of a generic chain).
5. Weak Form of Topological Laws We call a strong solution of a physical field problem, that which satisfies its mathematical model in terms of partial integro-differential equations supplemented by a set of boundary conditions [Oden, 1973]. Correspondingly, this mathematical model is called the strong formulation of the problem. Let us borrow this name and apply it to the differential formulation of topological laws. Hence, we shall call Equations (7) through (13) the strong form of the topological laws of electromagnetism, and Equations (70) through (71) the strong form of those of heat transfer. In the language of differential forms, these equations can be all rewritten as follows d~p = fl p + 1
(180)
where er and /3r+l are suitable differential forms representing the fields involved, and d is the exterior differential operator. We said that from the point of view of inverse limit systems, the operator d can be interpreted as the operator 600, that is, a collection of coboundary operators acting between the projections on the directed set Js of all the cell-complexes which subdivide the domain of the problem, of the fields represented by ~' and/3 p+ 1. Therefore, a strong topological statement such as Equation (180) can be interpreted as the collection of all the corresponding discrete topological statements (in terms of cochains and coboundary). Thus, Equation (180) is equivalent to 6AP(K) = B p+ I(K) VK ~ ~C
(181)
where AP(K) and B p+ I(K) are the cochains resulting from the projection of ~r and/~p+l on the cell complex K. Seen in this light, the weak and strong formulations of topological laws are different only in our considering the collection of topological statements as an assembly, or as a single entity. This approach applies also to the case of spread cells, that is, to the 15The collection of domains supporting the cells can be a foliation, or, in presence of singularities, a stratification [Abraham et al., 1988].
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enforcement of topological laws in terms of weighted integrals discussed above. Of course, the collection of spread cells must be wide enough so that practically all the conceivable topological statements will be enforced. Thus, selecting a suitable space ~ of weight functions, a statement such as
f~ c~P= fw tip + 1 Vw e ~U
(182)
w
(where, with some notational abuse, we have identified the weight function with the weighted domain of integration) "leaves nothing to be desired" [Bossavit, 1998b] from the point of view of the enforcement of the topological law expressed by Equation (180). In fact, it turns out that Equation (182) is actually a more comprehensive statement than Equation (180), since it is not disturbed by the presence of discontinuities in the field, which require instead the enforcement of separate interface conditions when adopting the strong formulation. Inspired by the language of functional analysis, we can call Equation (182) the weak formulation of the topological law. The equivalence between weak and strong formulations of topological laws no longer holds if, instead of the complete collection of cell-complexes that subdivide the domain, we consider one, or at most a few, cellcomplexes. This is the case of numerical methods, where only one mesh for each kind of orientation is built in the domain, and consequently only the topological statements corresponding to the actual meshes are enforced. Of course, since we are considering in the domain only the physical quantities associated with the geometric objects of the meshes, we cannot hope to enforce a wider set of topological equations than those that involve these quantities (which, however, are enforced exactly). In particular, if we build a field function defined on the whole domain, starting from the finite collection of global quantities defined on the complex and satisfying on it the corresponding topological law, we can expect those topological prescriptions not included in those enforced, to be violated [Bossavit, 1998a].
IV. METHODS
A. The Reference Discretization Strategy We have at this point all the elements to ascertain whether or not a discretization strategy complies with the tenets derived from an analysis of the mathematical structure of physical field theories. To provide a framework for the development of new methods that satisfy these requirements,
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and facilitate the comparison of these principles with those adopted by a number of popular numerical methods, we will now describe a reference discretization strategy, directly based on the ideas developed so far. Note that this reference strategy does not qualify as a complete numerical method, since the discretization of the constitutive relations is described in generic terms only. However, it will be clear that a whole class of methods complying with the analysis discussed so far can be obtained by combining the elements of the reference strategy. The reference discretization strategy is presented for the case of time-dependent electromagnetic problems, since at this point we know the structure of the factorization diagram for this theory, and it is in space-time that the analysis presented in the present work will come to full fruition. As a consequence of this choice of the problem for the reference strategy, the comparison that will follow will consider mostly time-domain methods, be they Finite Difference, Finite Volume, or Finite Element methods. Considering the wide scope of the ideas developed so far, however, a similar reference discretization strategy can be built for other kinds of field problems, in fact for those of any field theory admitting a factorization of its field equations into topological and constitutive equations. In summary, as the subject of the discretization, we consider, within a bounded space-time domain, a problem constituted by Maxwell's Equations (7), (8), (12), and (13) (or any of the integral formulations derived from them within the present work, in particular, the fully discrete form represented by Equations (56), (57), (61), and (62)), supplemented by a set of constitutive relations, for example Equations (14) through (16). To complete the definition of the problem, a set of initial and boundary conditions that make the problem well-posed will be assumed as given. Imposed currents and charges can also be specified as independent sources, for example, a term such as ,I v - p v is very common in problems deriving from particle accelerators design. 1. Domain Discretization
The space-time domain is discretized by the reference strategy using two dual oriented cell-complexes, which act as primary and secondary meshes. We assume that each mesh is obtained as a cartesian product of the elements of a cell-complex, which subdivides the problem domain in space, by those of a cell-complex discretizing the time interval for which a solution is sought. The more complex case of moving meshes, and the still more difficult case of generic space-time cell-complexes, could be contemplated as well, but entail a number of difficulties in the attribution of a physical meaning to the quantities and the deduction of suitable constitutive equations [Nguyen,
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FIGURE 33. Reference discretization of the domain in space 9Note that the orientation and index of each primary p-cell are used to index and orient the dual secondary (3 - p)-cell (the orientation of 0-cells and 3-cells is not represented) 9
1992], which we choose to avoid in this context. Remember, however, that the reference method can be extended to include these cases as well. Given this choice, we have for p = 0, 1, 2, 3 four collections of indexed primary p-cells {z~, 9 i = 1,..., np} in space. To each primary p-cell Zp /there corresponds a dual secondary (3 - p)-cell ~ _ p with the same index and the default orientation defined by that of the dual primary cell. The secondary cells also constitute, therefore, four collections of p-cells {~, i -- 1,..., hp = n3-p) (Figure 33). Note that to facilitate drawing, the 1- and 2- cells in Figure 33 and in Figures 35 and 36, are straight or planar, but this is not required by the definition of cell, on which the reference method is based. However, the use of planar and straight cells greatly simplifies the calculations, especially for what concerns the discretization of the constitutive relations. The actual construction of the two meshes is problem-dependent, since it must consider what kind of boundary conditions are specified (which determines what kind of cells are needed at the boundary), where the
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material discontinuities, if any, are located, and to what constitutive parameter they refer. This last point will become clearer after the discussion on constitutive relations discretization below. Generally speaking, one can start by defining a primary mesh that conforms to material and domain boundaries, and then construct the secondary cells by defining within each n-cell of the primary mesh (n is the dimension of the domain) a secondary 0-cell. The secondary mesh is then built starting from this 0-cell so as to make the two cell-complexes reciprocally duals. The actual position of each secondary 0-cell within its dual n-cell also depends on the problem and on the strategy adopted for the discretization of constitutive relations. In many cases, the position corresponding to the barycentre of the n-cell is a good choice, but, as will be hinted below, it is not always the optimal one. The domain in time is a time interval I subdivided by two dual cell-complexes. The primary one is constituted by two collections of indexed p-cells, with p = 0, 1. The 0-cells are time instants {t~, n = 1,..., N} indexed according to increasing time. To simplify the notation and facilitate the comparison with existing methods, the time interval going from t~ to t~+1 is indexed as t~ + 1/2. In time, to each primary cell t~ there corresponds a dual secondary cell 77_p, inheriting the index of its dual cell. Thus, the time interval 7'I goes from the time instant ~-1/2 to the time instant ~+ 1/2 (Figure 34). Primary space-time cells are obtained as cartesian products , and secondary ones as products z~iv x ~" zpi x tq, tq Note that the duality of the meshes applies in both space and time. This discretization supplies the oriented geometric objects needed to support the global physical quantities of electromagnetism, that is, QO, Q J, cb, Ce, 0d, oh, all/a, and ~v (defined in Equations (48) through (55)). However, the quantities actually appearing in the formulas of the reference strategy, are QJ, ~bb, ~be, ~,~, and Oh only. The association of a global quantity with a geometric object will be denoted by a pair of indexes, according to the following convention:
(183) ~e(~'l • t] + ~/~) = r
~/~
(184)
I//d(~ x ~3 + 1/2) __ Old,n+ 1/2
(185) (186)
QJ('g~ • 7]) = Qi,.
(187)
2. Topological Time-stepping It has been explained above how Faraday's law and Maxwell-Amp6re's law can be given a geometric interpretation in terms of global quantities
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primary
cells
t t
n
n+l/2
t
1
n+l
0...-......................"................"..0 "~' n - l / 2 \
v n ""
to
t
/
t
"~' t n+l/2
0
secondary cells FIGURE 34. Reference discretization of the domain in time 9
associated with a space-time cylinder (Figures 12 and 13). Within a domain discretized following the prescriptions of the previous subsection, we can apply this property to build a topological time-steppin9 procedure. Faraday's law is used to time-step 4? as follows. We build a space-time cylinder on a primary 2-cell r~ considered at the time instant t~. The resulting 2-cell r~ x tg is the first base of the cylinder. The boundary c~r~ of r~, considered during the time interval t'~+ 1/2 is a finite collection of 2-cells c?r~ x t] + 1/2 = 2;k[~2, r]] r] x t'l + 1/2 that constitutes the lateral surface of the cylinder. The cylinder is closed by the 2-cell ~ x t~ +1 (Figure 35). If we assume as known the primary global quantities at times t < tg +1, that is, qSib, and qSf,,n+l/2, we can calculate exactly 4)b,+1 from the topological equation ~(I )2 --- 0 3 [Equation (131)], which, isolating the unknown term, becomes
+,~.
~
~
~
(188)
k=l
where the actual sign of the second term of the right side depends on the default orientation assumed for the primal 2-cells to which a positive 4>e is associated. Using the representation [Equation (106)] of cochains as vectors
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FIGURE 35. Blown-up representation of the geometric objects and global physical quantities involved in topological time-stepping on the primary mesh. The (internal) orientation of the geometric objects is not represented.
of global physical quantities, and the definition [Equation (88)] of the incidence matrices, we can rewrite the topological time-stepping formula [Equation (188)] in matrix terms, as follows (i)b+ 1 -- (I)nb -- Oz,l(I)n+ e 1/2
(189)
where we assume the default orientation, which gives to the last term a minus sign. An analogous procedure holds for the time-stepping of Od by means of Maxwell-Amp6re's law 6CIJ 2 = 0 3 on the secondary mesh (Figure 36). The result is
l[lidn+ 1/2 = I[lidn
1/2 .ql_ 2 [ ' ~ ' ,~k] i//h,n nt_ k
Q{,,
(190)
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FIGURE 36. Blown-up representation of the geometric objects and global physical quantities involved in topological time-stepping on the secondary mesh. The (external) orientation of the geometric objects is not represented.
where QI,,, is the charge associated by the electric current flowing through "~ during the time interval 7]. With a suitable choice of default orientations, the matricial representation of Equation (190) corresponding to Equation (189) is " d 1/2 ~-- kiln" d 1/2 + I)2,1kIIn "~h -- Q~ kiln+
(191)
If we consider the collection {q~ibo} given as part of the initial conditions, we can use Equation (188) to start a time-stepping for ~bb, provided the set of values {4~f,,a/2} is known. Of course, we can't expect these values to be also given as initial conditions. We can, however, assume the set of values {~//id1/2} to be given as initial conditions. Hence, we can derive {r from {g'/{1/2} by means of a discrete constitutive link F,;-,, and advance in time
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FIGURE 37. The two half-time steps of the reference method. Topological time-stepping is applied on each side of the diagram, to update q5b and ~a, respectively. The discrete constitutive links supply the quantities required by the time-stepping formulas, that is, ~be for the updating of 4~b, and ~h and QJ for the updating of ~a.
in this way q~b from {4~,b0} to {~b,bl}. At this point we know {qSi,1} b and we can derive {Oihl} from it by means of a discrete constitutive link F~,-,, and {Q].I} from {q5~,~/2} and {q5~,3/2} or, better, indirectly from {O,a,/2} and {~,3/2} by means of a constitutive link Fa, e-I , which includes the action of the constitutive link f~-i and of f~. This allows the determination of {Oia3/2} by time-stepping {~al/2 } using Equation (190), and so on (Figure 37). In matricial representation, for a generic time step n, this corresponds to O~ +1
b -- D2,1 --- (I)n
"a 1/2 -- kr~n-1/2 ~a kiln+
F~-I(~ a + 1/2)
"[- 6 2 , 1 F ~ - ~ ( ~ ) -- Fr _ , ( ~ a )
(192) (193)
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where in the term F~-,(~ a) the cochain has no time-step subscript, as a reminder that both ~ , + ~/2 and ~ , _ ~/2 are involved in the link. The topological time-stepping formulas [Equations (188) and (190)] are based on two of Maxwell's four equations. We will now prove that in adopting the time-stepping scheme thus described, we won't need to explicitly enforce the other pair of Maxwell's equations. As usual for numerical methods devoted to time-domain electromagnetic problems, it will suffice to show that, if these equations are satisfied at a given time instant, they remain so after the execution of a time-step. Consider first Gauss's magnetic law. This asserts the vanishing of magnetic flux associated with the boundary of any 3-cell z3 at any time instant tg. To simplify the notation, we denote with @, the magnetic flux cochain at time tg, thus avoiding the use of products of space-like and time-like geometric objects in the formulas. With this provision, Gauss's law for a particular 3-cell z~ considered at time t~ reads (194)
(~b(~.~ X /-~) = (~'C:~, (I)n) = O
where, in the middle term, we have represented the quantity in terms of a chain-cochain pairing. We must now show that from the validity of Equation (194) and the application of the time-stepping formula [Equation (188)], there follows the validity of Gauss's law at time t~ + 1. Substituting in Equation (194) the expression in Equation (96) of the boundary in terms of incidence numbers, we have
= ( ~ [r~, r~]r~, 0 . ' ) = ~ [r~, r~]4~ib,. = 0 \ i / i
(195)
The same substitution, applied to the expression of ~bb at time t~+ 1, gives ~bb(cOr~' x t~ +') = y' [r~, z~]~bib.+,
(196)
i
Substituting the time-stepping formula of Equation (188) in the right side of Equation (196) we obtain S ITS, T/2]r i
1 = Z [T~, Z~]~bn Jr- S [r~, qT~] Z [r~, "/Tk]r i i k
1/2
(197)
Rearranging the terms, we have
Z [z~, z~]4~ib,.+, = ~ [r~, r~] 4~,b,.--+ Z ~ [r~', z~] [r~, z]] 4ff,,.+,/2 i
i
i k
(198)
The first term of the right side of Equation (198) vanishes, since we have assumed Equation (195) to hold true; the second term vanishes in virtue of
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the relation [Equation (90)] holding among incidence numbers. Hence, remembering Equation (196), we finally have ~bb(ctz~, • t~+ 1) = 0
(199)
This proves that if Gauss's magnetic law is satisfied at time t~, then, applying the topological time-stepping formula, it is also satisfied at time t~+ 1. From Equation (198) there follows a more general conclusion, namely, that following the execution of topological time-stepping, the amount of violation of Gauss's magnetic law, if any, does not change. In other words, the topological time-stepping on ~bb automatically enforces the law of magnetic charge conservation. In the case of Gauss's electric law the balance to be enforced is that between the electric charge associated with the 3-cells and the electric flux through its boundary. Suppose that at time ~-1/2 there is a charge Qp,,,_ 1/2 associated with the 3-cell ~ , and that the following relation holds t/Ja(ct~'~ • t~- 1/2) = Qp,,,_ 1/2
(200)
Repeating the steps of the above proof, but using instead the time-stepping formula of Equation (190), we obtain
~ja(c~.~ • 7~+ 1/2) = Qp.,._ 1/2 ___~ ['~, "~2] Qi,.
(201)
i
This shows that after topological time-stepping, the electric flux associated with the boundary of ~ may have changed. But this change is consistent with the law of charge conservation, since the new term on the right side of Equation (201) is the result of the electric current flowing through the boundary of the cell during the time interval t]. Hence, the topological time-stepping on Od enforces automatically the law of electric charge conservation, and preserves the violation of Gauss's electric law, if any. Note how the realization of the space-time nature of topological laws suggests the adoption of a uniquely determined time-stepping procedure on each side of the factorization diagram of physical quantities, an observation that is true for any theory admitting such a factorization diagram. It is clear that we are revealing here the roots of the adoption, within many numerical methods for partial differential equations, of a leapfrog time-stepping procedure based on two half-time steps, a choice that, in the absence of the justification given in the present work, which is based on the analysis of the structure of physical theories, is often considered as an oddity, justified only by the good results it offers. The limits that the univocity of the topological time-stepping process puts on the form of the complete time-stepping are not as severe as it might
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appear at first sight. It is indeed true that the topological time-stepping formulas (35) and (36), are based on a topological law applied to a single, space-time 3-cell, and, therefore, that each newly calculated value directly depends only on quantities associated with the cell itself and with its boundary. The complete time-stepping operator includes, however, in addition to the topological relation, at least one discrete constitutive operator. This constitutive operator links the quantities directly involved in the topological time-stepping formula to other quantities. The constitutive link, therefore, allows the extension both in space and time of the dependence of the newly calculated value on quantities associated with cells other than those for which the topological law is enforced. Thus, the newly calculated values associated with a given cell can be made to depend on quantities associated with cells of a generic neighborhood of that cell in space, and extending in time deeper into the past than the single time-step considered by the topological relation, or on other quantities for the time-instant at which the new quantity is calculated. This can be expressed rewriting the particular time-stepping formulas of Equations (192) and (193), as follows (202)
+ 1 - - (I)n - - D 2 . 1 F e - ' ( ~ a ) - '~ 1/2 - - kIJn~'~ 1/2 -Jr- 6 2 , 1 F /1 _ , ( + b ) kIJn+
_
F~. ,~~-, (~"~)
(203)
that is, removing the time subscript from the cochains entering the discrete constitutive links, in order to emphasize the involvement of the whole space-time cochain in the link. Observe that if the expression of the discrete constitutive operators is explicitly given, by actually substituting them in the topological time-stepping formulas, we can make them depend on only two kinds of variables, in this case 4)b and ~d (but other pairs of variables can be selected to appear in the formulas, applying differently the constitutive links). In summary, there is a possibility of building a variety of time-stepping procedures complying with the adoption of a topological time-stepping operator. Given the variety of discrete constitutive links that can be built, and the uniqueness of the topological time-stepping links, one could, therefore, conceive a numerical package offering the choice between different discretization strategies for constitutive relations, to be combined with the unique discretization of topological laws based on the coboundary operator. Note finally that we can expect problems in trying to use a weighted residual approach to build a topological time-stepping procedure, for the geometric ideas upon which we have based the topological time-stepping procedure cannot be easily extended to spread cells.
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3. Strategies for Constitutive Relations Discretization
The task of constitutive relations discretization consists in the determination of a link between cochains, which approximates the local constitutive equation and, with it, the ideal material behavior it represents. There are many possible approaches to this task, and from this point of view the three cases presented below make no pretense of being exhaustive. On the other hand, since many numerical methods do not consider explicitly the particular problem represented by the discretization of constitutive relations and perform this task in a manner that appears at most as some form of educated guessing, we will try at least to present discretization procedures for constitutive relations that can be applied systematically. Our inspiration, as usual, comes from existing numerical methods. We will first consider one of the simplest and most intuitive approaches that qualify as a systematic technique for the discretization of constitutive relations; then two more sophisticated classes of strategies are presented. Remark: The discretization of constitutive relations is often referred to as the discretization of the Hodge star operator 9 [Tarhasaari et al., 1998; Teixeira and Chew, 1999] (see Bamberg and Sternberg [1988] for a formal definition of its action). Considering the great variety of possible constitutive links, this point of view appears too restrictive. The Hodge star operator institutes indeed a one-to-one correspondence between ordinary p-forms ~P and twisted (n - p ) - f o r m s fl"-P defined on an n-dimensional manifold. We can represent this relation as follows
fl"-P = 9 ~P
(204)
It is apparent from Equation (204) that adopting the representation of fields as differential forms, the Hodge operator can play the part of a constitutive link (provided we include within it the required material parameters). However, in this role, the Hodge operator is a mathematical model for the behavior of a particular class of ideal materials (and when it is considered merely in mathematical terms, i.e., without the intervention of material parameters, not even that), and cannot be considered as a model for all material behaviors. It is the fact that the Hodge star operator constitutes the traditional bridge between ordinary and twisted forms that tempts us to consider it the constitutive operator. That things are not so can be seen by considering that constitutive equations can be mathematically much more complex than the simple correspondence brought about by the Hodge operator (see, for example, the discussion in Post [1997]). Of course, since the transition from ordinary to twisted differential forms (or vice versa) is implied by the constitutive links, the Hodge operator or something analog-
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ous, capable of "crossing the bridge" in the factorization diagram, will be actually required, but typically only as a part of the complete constitutive link. In other words every operator linking two differential forms that represent two fields plays the role of the constitutive relation of a particular ideal material (perhaps a nonphysical one, as in the case of materials that form the so-called Perfectly Matched Layer, used for the implementation of absorbing boundary conditions [Berenger, 1994; Teixeira and Chew, 1998]). However, contrary to what happens with the topological equations, no single operator can claim a privileged role as constitutive operator. a. Discretization Strategy 1: Global Application of Local Constitutive Statements While introducing the idea of constitutive links, we hinted at the fact that a local constitutive equation of the kind D = eE holds true also in a macroscopic space-time region, provided that the fields are uniform in space and constant in time, and that the material is homogeneous. In this case, if the surface S to which the electric flux is associated is planar, and orthogonal to the straight line segment L to which the voltage is associated, we can write
~d
~be = e-S LI
(205)
where I is the time interval during which the voltage is considered and, with some notational abuse, we have identified the symbols of the geometric object with the value of their extension. The uniformity conditions upon which the transition from the local statement to the global one is based are admittedly quite severe. Consequently these requirements are not satisfied in the great majority of cases, and equations such as Equation (205) are, therefore, only rough approximations of the actual relation holding between the global variables. Nonetheless, many numerical methods adopt this approach more or less explicitly, in order to obtain a discrete version of the constitutive links. The rationale behind this choice is that decreasing the maximum space and time discretization steps, the uniformity hypothesis is approached more and more closely and Equation (205) becomes an acceptable discrete approximation of D = eE. Note that in addition to uniformity there is also a requirement of geometric regularity and orthogonality of the geometric objects and, therefore, of the discretization meshes. Hence, this approach will require meshes with dual, orthogonal cells, for example the regular orthogonal grids of FDTD, or the more general Delaunay-Voronoi meshes [Guibas and Stolfi, 1985]. This last requirement, however, can be usually
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relaxed, paying the price for a more complex evaluation of the terms appearing in the link, to take care, for example, of the angles between the cells or their curvature. In summary, applying consistently this simple discretization technique to the local expression of all the constitutive relations appearing in a problerri, one obtains a series of links between cochains, which are the required discrete constitutive links. Since this approach can give only very crude approximations of the actual constitutive relations, it is acceptable only if the fields do not vary rapidly in space and time, or if the recourse to a very fine mesh can be accepted. To find, for the constitutive relations, a more accurate discrete approximation than the one just presented, it is clear that we must use more extensively the information represented by the local constitutive equations. We will consider, in the next sections, two approaches based on the preliminary reconstruction--based on the corresponding c o c h a i n s - - o f one or both of the field functions appearing in the constitutive equation written in local form. b. Discretization Strategy 2: Field Function Reconstruction and Projection The method considered in the present section requires the reconstruction of only one of the field functions appearing in the constitutive equation in local form. An example will clarify the actual workings of the strategy. Suppose that we want to discretize the following constitutive equation: B = fu (H)
(206)
As usual, to simplify the notation we represent the field functions using the traditional tools of vector calculus, even though a formulation in terms of differential forms would be more appropriate (see Teixeira and Chew [1999] for a description of the strategy both in differential forms and vector calculus language). In the discrete setting of the reference strategy, the field functions B and H appearing in Equation (206) do not belong to the problem's variables, and we have instead the magnetic flux cochain ~b and the electric flux cochain ~,h, which at the end of the process must be linked by the relation ~ b = F~(~h). In order to use the information constituted by Equation (206) we proceed by deriving from the cochain ~h field function H. To this end, we select a reconstruction operator Rh giving for each cochain ~h a field function H, as follows n = R h(~' h)
(207)
Note that the reconstruction in Equation (207) starts from space-time global quantities, and, therefore, the reconstructed field is intended as given in space-time also, as a function H(r, t). We can now apply to H the local constitutive link Equation (206), obtaining the field function B. We must
100
CLAUDIO MATTIUSSI
FIGURE 38. The reconstruction-projection method for the discretization of constitutive relations. Given the starting cochain, an approximation of the corresponding field function is determined by means of a reconstruction operator R,. The result is then subjected to the action of one or more local constitutive operators f , . The resulting field function is finally projected on the cell-complex by means of an operator P*, thus recovering a cochain.
finally return to cochains, and this we do by means of a projection operator pb, which produces a cochain ~b for each field function B, as follows: ~ b = pb(B )
(208)
The composition of the operators [Equations (206), (207), and (208)], gives the desired discrete constitutive link C
Fu(~ h) -- Ph( fu(Rh(CFh)) )
(209)
The same approach applies to the discretization of the electrostatic link, and of any other constitutive relation, of which the local expression corresponding to Equation (206) is known (Figure 38). A natural joint requirement for reconstruction and projection operators is that for every cochain c p the following relation holds true P*(R,(cr)) = c ~'
(210)
N U M E R I C A L M E T H O D S F O R PHYSICAL F I E L D P R O B L E M S
101
that is, by projecting back the reconstructed field one must obtain the original cochain. This means that the combined operator P ' R , must be the identity operator. Note, however, that this is not true in general for the operator R,P*, that is, due to the limitations of the reconstruction operator, by projecting a generic field and then reconstructing it one typically does not obtain the original field [Tarhasaari et al., 1998]. Note that in Equations (207) and (208) we added to the symbols R and P of the reconstruction and projection operators an index, which refers to the field function involved in the process. This was done to serve as a reminder that the operators must comply with the association of physical quantities with oriented geometric objects, so that each operator will be "tailored" to the actual nature of the fields it is called to operate on. In particular, the reconstruction operator operates on p-cochains, and produces p-fields that must be compatible with the original cochain. It is clear, therefore, that the proper selection of the reconstruction operator is instrumental in the attainment of a good discrete solution. For this reason, a separate section below is devoted to the discussion of some actual reconstruction operators. Using the just derived representation of the discrete constitutive operators, we can rewrite the generic time-stepping formulas (202) and (203) for this particular discretization strategy, as follows r +~= r b + D~,~ pe (L -1 (R.(q'")))
~d 1/2 -- kIJn~d 1/2 -~- I~2,1 kIJn+
ph(f_,(Rb(C,b)) ) + W(f~(L_,(Rd(~d)))
(211) (212)
With the reconstruction and projection operators appearing in Equations (211) and (212), the reference discretization strategy becomes a class of numerical method complying with the reference discretization strategy. Let us analyze in general terms where the discretization error might enter these class of methods. The strategy first requires the reconstruction of the field function, starting from a cochain. This opens the first door to possible errors since we can't expect the true solution to be in the range of our reconstruction operator. Remembering what has been said about limit systems and the concept of field as a collection of its manifestations in terms of cochains on the directed set of all the cell-complexes that subdivide the domain, this can be interpreted by saying that a single cochain alone cannot determine the field it derives from. More precisely, given a cell-complex and a cochain defined on it, there would be in general an infinite number of fields that admit that cochain as its projection on that complex. The choice of the reconstruction operator corresponds, therefore, to the selection of a particular field in the multiplicity of fields that are compatible with the discrete image we start from.
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CLAUDIO MATTIUSSI
After the reconstruction step and the application of the local constitutive operator, we project the resulting field function onto the cell-complex, obtaining a cochain, and we impose on it the topological equation. Since the cell-complex is finite, it follows that we are enforcing only a finite subset of all the possible topological relations implied by the corresponding topological laws. This gives more freedom to the solution than what was implied by the original physical field problem. It is the combination of these two processes that gives rise to the discretization error that we will finally find in the ideal solution of the discrete problem. This double nature of the discretization error was lucidly analyzed by Schroeder and Wolff [1994]. The reconstruction-projection strategy for the discretization of constitutive relations, and the corresponding error analysis, was given a formal treatment based on the concepts of the theory of categories [Tarhasaari et al., 1998]. In that context, the name Whitney functor and De Rham functor was suggested for the reconstruction and projection operators, respectively. Let us add some further comment regarding the properties of the projection operator. Speaking of limit systems, we said that a field can be thought of as a collection of cochains, each of which is a projection of the field on a particular cell-complex. Adopting the natural representation of a cochain as a vector of global physical quantities associated with cells, the operation of projection amounts, therefore, to the evaluation of these global quantities on the p-cells of the complex, where p and the complex orientation must suit the nature of the field. In practice, this evaluation corresponds usually to the integration of the field function on these p-cells. This fact, combined with the fact that the reconstruction operator performs an approximation of the field function, opens the door to some optimization opportunities. We know from the theory of approximation that the reconstructed function typically has a set of loci where the approximation is of a higher order than in generic points of the domain. Therefore, by making the cells where the projection is performed coincide with those loci, we can obtain a higher accuracy in the resulting discrete constitutive equation. This means that the accuracy of the results can benefit from the proper selection and placement of the primary and secondary meshes. In particular, it can be shown that the choice of two suitably placed dual grids can be desirable also in this respect [Mattiussi, 1997]. For low-order polynomial reconstructions on regular cells, the center of the cells is usually the optimal location for the dual object. This extends to the time-stepping procedure, where it suggests the placement of secondary time instants in the middle of primary time intervals. Note that the approach used by the reconstruction-projection strategy to discretize the constitutive relations gives rise to discrete links where the value of the resulting cochain on each cell depends on the values taken by
N U M E R I C A L M E T H O D S FOR PHYSICAL F I E L D PROBLEMS
103
the cochain one starts with, on many cells, potentially on all those entering the reconstruction process. In order to preserve the sparsity of the matrices appearing in the system of algebraic equations, the reconstruction process is usually performed locally, so that the value of the reconstructed field function on each point depends only on the values taken by the original cochain on the cells of a sufficiently small neighborhood of the point. In particular, the simple discretization strategy described in the previous subsection is a particular case of the strategy described here. In that case the reconstruction operator works locally on a single cell, giving a uniform field, which is projected onto the dual cell. c. Discretization Strategy 3: Error-Based Discretization
There is another approach to the discretization of constitutive relations, which is based on the reconstruction of field functions. Let us describe the workings of this strategy using the same example of the previous strategy. As before, we assume as known the electric flux cochain ~h and we want to determine the magnetic flux cochain (I)b. Contrary to the previous case, however, we apply a reconstruction operator to both cochains as follows: H = Rh(tP h)
(213)
B = R b(~b)
(214)
Since the cochain (I)b is the unknown term of the discrete link, the reconstruction of B is made in formal terms only. Ideally, the relation holding between B and H is the local constitutive equation B = f,(H). F r o m the discussion of the previous subsection we know t h a t - - b o t h fields being obtained by reconstruction from cochains - - these fields are forced to be in the range of the reconstruction operators. Consequently, we cannot expect the local constitutive equation to be satisfied exactly by the reconstructed fields. We, therefore, define an error density function in the domain, which we denote with
~(B, H)
(215)
which is intended to give a local estimate of the amount of the violation of the constitutive link. As the minimal set of requirements for this scalar function, which measures the local constitutive error, we ask it to be always positive, and to vanish only for B and H satisfying B = f,(H). The actual definition of the function ~ is a nontrivial task, which depends on the problem and on its constitutive relations. We will not consider in detail this subject here, assuming this function as given. For this important topic the reader is referred to the literature, in particular to that dealing with
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C L A U D I O MATTIUSSI
complementary variational techniques, which appear especially suited to the determination of physically significant local error functions linked to local and global energy estimates [-Rikabi et al., 1988; Oden, 1973; Penman, 1988; Albanese and Rubinacci, 1998; Marmin et al., 1998; Remacle et al., 1998]. Substituting the reconstruction operators [Equations (213) and (214)] in the error function [Equation (215)], we obtain the local error function in terms of the cochains ~h and ~b ~(B, H) = ~(Rb((I)b), Rh(~h))
(216)
Integrating ~ on a space-time domain ~ we obtain a global error functional t" ~((i)b, ~h) = .Ion ~(Rb((I)b)' Rh(Wh)) Since the cochain tTph is known, we can determine the optimal cochain ~_b by means of the following optimization problem ~_b = ~b. min~ (~b, ~h) ~b
(217)
Fu(~h)
This procedure implicitly defines a link of the kind ~_b= and, therefore, establishes a discrete constitutive relation approximating the local constitutive relation. Note that this approach to the discretization of a constitutive relation puts the two fields linked by the equation on a more equal level than the previous strategy. There is, of course, still a direction of the link, going from the known cochain to the unknown one, but, in terms of the coefficients of the cochains involved, the link is no longer many-to-one but many-to-many. From a physical point of view this appears sound, considering that a constitutive relation should not be considered a cause-effect relationship where a field is given, which fully determines another field, but instead must be viewed as a constraint, which co-determines both fields. Note also that, generally speaking, the minimization problem [Equation (217)] takes place in space-time, and not in space only. In fact, in a time-dependent problem, a minimization procedure must be performed at each step, for example, on the space-time domain constituted by the cartesian product of the domain D in space multiplied by the time step At as follows ~ (R b((I)b), Rh (~h)) The necessity of running an optimization problem at each time step can greatly increase the computational cost of this strategy. In fact, the errorbased approach was applied in the past mainly to static or quasistatic problems, or to frequency-domain ones I-Albanese and Rubinacci, 1993].
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
105
This is also due to the fact that the required theoretical analyses and error functions were first given for these cases. However, the development of computing machines can quickly make attractive this kind of approach in alternative to the reconstruction-projection strategy also for time-dependent problems [Albanese et al., 1994; Albanese and Rubinacci, 1998].
4. Edge Elements and Field Reconstruction The discretization strategies for constitutive relations that we have presented require the reconstruction of field functions based on the unknowns of the discrete formulation of the problem. This appears at first as a familiar problem with function approximation. However, in our case, the starting data are cochains, that is, collections of global values on cells, not nodal samples of field functions. In other words, instead of a traditional nodebased approximation problem, we must consider a problem of cochain-based field function approximation [Mattiussi, 1997; Mattiussi, 1998] (Figure 39). The two concepts coincide only for the case of quantities associated with points, which, for theories having scalar global values, are represented in the continuous case by 0-forms (i.e. is, by scalar field functions), and in the discrete case by scalar-valued 0-cochains. Working only with unknowns associated with points, one can easily overlook the fact that the reconstruction is actually based on cochains, and this is what happens, for example, with a potential formulation of electrostatics. However, already in magnetostatics, one is faced with the fact that neither the fields nor the vector
/
\
/
\
C
-\
/
\
/ I \~ 9
~ v -18
~''~"'--~ C
19
FIGURE 39. Nodal-based field function approximation (left) is based on a set of local scalar or vector values defined on a grid of points. Cochain-based field function approximation (right) takes instead as its starting point a p-cochain, that is, a set of global values associated with the oriented p-cells of the mesh. Here the case of ordinary 1-cochains on 2-dimensional domains is considered.
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CLAUDIO MATTIUSSI
potential are associated with points. To use the traditional node-based tools of approximation theory in this case one is, therefore, forced to ignore the correct association of physical quantities with geometric objects, and consequently also to abandon any hope of complying with the structure of the field problem. The alternative is the introduction of new approximation tools tailored to the characteristics of cochains. In this sense it can be said that one needs to consider at least 3D magnetostatics to start appreciating the true nature of the task constituted by the discretization of an electromagnetic problem. In summary, while an ordinary approximation problem asks: "find on a given domain a scalar-valued or vector-valued function which approximates the data constituted by local scalar or vector values defined on a set of points," our approximation problem states: "find on a given domain a p-form that approximates the data constituted by global values associated with the p-cells of a cell-complex." In other words, we are requiring of the reconstructed p-form to have the given cochain as its projection onto the complex. A traditional approach to the solution of approximation problems within numerical methods is the selection of a set of shape functions. In our case, these are suitable forms crf(r), that is, p-forms with the correct kind orientation for the p-cochain one starts with, which can be used as a basis for the reconstruction, for example, in terms of a linear combination of them, as in ~oP(r) = ~ aioP(r) i
(218)
The reconstruction must, of course, be uniquely determined (and, in fact, it must be a well-conditioned operation), and this is reflected in independence requirements of the shape functions. If instead of differential forms, we choose to work with the traditional tools of vector calculus, the entity to be approximated is a scalar or vector function, that is, a p-field defined in the domain. Correspondingly the forms o'~'(r) become field functions s~'(r) or s~'(r). From now on we will speak generically of shape functions, including in this definition differential forms, scalar and vector functions. Generally speaking, the shape functions for an approximation problem are defined globally, that is, they are nonzero on the whole domain. In numerical methods for field problems it is often preferable to define shape functions of a local nature, that is, functions that are nonzero only within the domain constituted by a small number of adjacent cells. A class of shape function, which complies with all the requirements listed so far, is that of the so-called edge elements. These are shape functions, which were introduced some twenty years ago in finite element practice [Jin, 1993; Albanese and Rubinacci, 1998]. Edge elements are usually defined in terms of the kind of
N U M E R I C A L M E T H O D S FOR PHYSICAL F I E L D PROBLEMS
107
interelement discontinuities they permit. Here, we will offer instead the following definition: an edge element is a shape function aP defined in a domain subdivided by a cell-complex, whose projection on the cell-complex is an elementary p-cochain, that is, a cochain whose value is one on a particular p-cell z ei and is zero on all other p-cells of the cell-complex. In formulas
'~ cr~ =
if i 4: j
note that Equation (219) is a natural extension to generic geometric objects of the requirement traditionally imposed on the nodes of scalar shape functions. If the shapes functions in the reconstruction Equation (218) satisfy Equation (219), they satisfy automatically also the property Equation (210), that is, we have P ' R , = I. To comply with the requirements of numerical methods, we ask also of this shape function to be nonzero only on a small neighborhood of zl,. We will call such a shape function an ordinary or twisted (depending on the orientation of the corresponding cochain) p-edge element, to emphasize the correspondence to a particular oriented geometric object. The definition of edge elements given above is intended as a unifying definition in terms of the role they play in the discretization process, that of cochain-based field function approximation (their possible role as weight functions is discussed later, in the context of finite elements methods). Paralleling the reasons behind the introduction of the reference discretization strategy, this definition of the edge elements is not intended to give a sterile classification, but rather to help in testing existing elements for their consistency with this role, to be a guide for the development of new elements, and to assist in extending the application of edge elements to new fields. Our definition of edge elements might seem strange to edge elements practitioners also for the fact that they are used to taking as their starting point the averaged c o m p o n e n t s of the field to be reconstructed, tangent or normal to p-cells. A bit of reflection, however, reveals that the two ideas are perfectly equivalent, since multiplying the averaged field component by the extension of the cell one obtains the global value associated with the cell, whereas the fact that only the averaged tangential or normal component (to accommodate internal and external orientation, respectively) is considered assures that the field quantities contain no more information than the global value (Figure 40). From our definition of edge elements it follows that given a cochain e p on a cell-complex K and a set of edge elements {a~}, one for each p-cell of K, we can construct a field function in the domain ]KI as a linear combination
CLAUDIO MATTIUSSI
108
C
2
C1 V1
u 2
FIGURE 40. Edge elements are usually considered as based on the averaged components of the field tangent to the cells or normal to them (left column). This is, however, equivalent to considering the corresponding global values associated with internally or externally oriented cells, respectively (right column). Edge elements for two-dimensional problems are considered here.
of the kind (218), but this time with the coefficients c i that are the values taken by the cochain to be reconstructed, on the p-cells of the complex, that is, the vector that represents the cochain with respect to the natural basis for cochains [Equation (106)]. The simple requirement of having an elementary cochain as their projection does not determine uniquely edge elements. One can indeed find a multitude of shape functions that comply with this requirement, and in particular with Equation (219). For this reason, one tries, in selecting edge elements for a problem, to satisfy other properties also, in particular those related to the accuracy of the reconstruction and, therefore, of the computation. These include, for example, the presence of the polynomial terms up to a given order in the reconstructed functions or in a transformation thereof [Sun et al., 1995]. Note that in some cases a certain number of
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
109
missing terms in the reconstructed function can be dispensed of with a proper placement of the meshes [Mattiussi, 1997]. In this quest for a higher-order edge element, it may happen that one ends up defining shape functions that resemble edge elements but, according to the definition given above, are not. This is the case of a number of the so-called vector elements that have been proposed to improve the behavior of the first generation of edge elements [Cendes, 1991; Sun et al., 1995]. Let us follow the path that leads to the introduction of these elements. We consider 1-edge elements (i.e., elements whose projection are 1-cochains) for two-dimensional problems, and we represent them with the degrees of freedom they associate with a triangle (Figure 40). It is apparent that edge elements of this kind permit the reconstruction of 1-fields on the primary and secondary mesh. They are characterized, however, by a small number of degrees of freedom, and, therefore, by a small number of terms in the approximating polynomials. This translates in a slow rate of convergence for the methods where they are employed [Sun et al., 1995]. To circumvent this problem, it is natural to increase the number of degrees of freedom associated with each edge element. Figure 41 shows the result, in terms of degrees of freedom, for a popular vector element derived following this idea. It is apparent that this elements mixes internal and external orientation, and associates multiple quantities with a single geometric object. Thus, reconstruction based on this kind of elements violates two of the fundamental principles of the association of physical quantities with
u 2
V
~V3 \
/
/
~
8
4
FIGURE41. Anomalous edge elements mix internal and external orientation, and associate multiple quantities with a single geometric object, thus violating the principles of the association of physical quantities with geometric objects.
110
CLAUDIO MATTIUSSI
C 2
C
4
+6 5
FIGURE 42. Anomalous edge elements can be brought back to conformity with the prescriptions deriving from the structural analysis of physical field theories, by introducing additional geometric objects.
geometric objects. Since they do not comply with our definition of edge elements, we will call these kind of elements, anomalous edye elements. Anomalous edge elements show that not every non-nodal-based shape function is an edge element. There is no doubt, however, that the introduction of such elements was dictated by the necessity to overcome some real problems. Let us, therefore, try to bring them back within our definition of edge element. To this end, we must introduce additional geometric objects. Assuming that the quantity under consideration is associated with internally oriented cells, the new geometric objects are primary 1-cells, as shown in Figure 42. In this way each physical quantity is associated with a distinct geometric object, and the kind of orientation is the same for all the cells intervening in the reconstruction. Figure 42 shows also that a given p-edge element can be defined on the p-cells of a domain, which includes many n-cells of the mesh (in an n-dimensional problem). In fact, this is the easiest way to raise the number of terms in the resulting interpolating polynomial. It appears that there is, therefore, an additional discretization structure for the geometry, linked to the operation of reconstruction suggested by the discretization of constitutive relations. These additional discretization entities we call the elements, are the domains on which separate approximation problems are solved to reconstruct the field function starting from the cochain coefficients. In other words, only the coefficients corresponding to cells included in the element are used to build the approximation, which holds true within the element. The corresponding structure we call the
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
111
FIGURE 43. The recourse to field reconstruction for the discretization of the constitutive relations requires the definition of an additional geometric structure, in addition to the primary and secondary mesh, namely an element mesh for each field to be reconstructed. Usually each object of the new mesh is obtained as the union of cells of the primary or secondary mesh. This, however, is not mandatory.
element mesh (Figure 43). Usually each element is constituted by the union of a small number of n-cells of the mesh where the reconstruction takes place. This is, however, not mandatory, and one could easily conceive of a separately defined element mesh to accommodate this process or, as is the case of meshless methods [Belytschko et al., 1996], no element mesh at all. Finally, some scattered comments on edge elements and the operation of reconstruction in general. First, we should mention that edge elements alone do not guarantee that a discretization complies with the results of the analysis of the structure of physical theories. Given a cochain, edge elements reconstruct a field, which complies with that cochain, in the sense that the latter is a projection of the former. However, if the cochain we start with is a nonphysical one, in that, for example, it does not satisfy the topological laws of our theory, we cannot ask the reconstructed field to do so. Hence, only a proper formulation of the field problem, along with the use of edge elements in the reconstruction of field functions, guarantees the physical soundness of the solution [Mur, 1994]. Next, note that within the approach presented here, shape function is not used to obtain a continuous field defined on the whole domain, but only as a step in the realization of the discretization strategies for constitutive relations. F r o m this point of view, they are a tool used temporarily in a phase of the discretization process after which they are discarded. Of course, one must not be careless in using this
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CLAUDIO MATTIUSSI
tool. In particular, one must consider the fact that the discontinuities in the properties of materials usually produce corresponding discontinuities in the field functions, or in their derivatives. Therefore, to properly approximate the constitutive equations in elements containing material discontinuities, one sees here the necessity--for the first t i m e - - o f taking into account these material discontinuities, and to make them coincide with element boundaries, so that discontinuities of the fields or of their derivatives can be modelled by the reconstructed field. We emphasize "for the first time" because the discrete rendering of topological laws, as presented above, is not disturbed by the presence of material discontinuities, since topological laws do not depend on material parameters. In fact, when dealing with global quantities only, the very concept of field continuity is meaningless. Note also that the reconstruction is instrumental to the constitutive relations discretization, which implies that we do not ask the reconstructed fields to satisfy the topological laws (in local, differential, form), since these are imposed only (in global form) at the cell-complex level. B. Finite Difference Methods
We now deal with the comparison of existing methods with the reference discretization strategy detailed above, starting with Finite Difference methods (FD). We should mention in advance that in presenting the methods the references typically cited are not founding papers but preferably survey works including extensive bibliographies. The classical FD approach to the discretization of field problems is based on the use of finite difference formulas to approximate locally the derivatives entering the expression of differential operators. A structured grid of points is defined first--usually a very regular o n e - - a n d a local field quantity is attached to each point. Then, in each of these points the differential operators appearing in the problem's equations are given a discrete expression by means of the above-mentioned finite difference formulas. Given the absence of any reference to the association of physical quantities with geometric objects other than points, one can hardly expect such an approach to give results consistent with the analysis developed thus far. This is indeed the case for the first attempts to give an FD formulation for electromagnetic problems. These resulted in methods that have practically nothing in common with the reference method developed above. A first notable exception is constituted by the well-known Finite Difference-Time Domain method, the analysis of which offers some interesting results. 1. The Finite Difference-Time Domain Method
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113
Consider an electromagnetic problem with constitutive equations B =/~H and D --- eE, and, for simplicity, the absence of electric losses. To solve this problem, the Finite Difference-Time Domain (FDTD) method starts by defining within the space-time domain of the problem two dual orthogonal cartesian grids. These subdivide the space domain in parallelepipeds having size Ax, Ay, and Az. The time domain is subdivided in time steps of size At. The primary and secondary grid are staggered by a half step in both space and time. To preserve the distinction emphasized thus far between primary and secondary geometric objects, and the corresponding one between the related quantities, we will write Ax, Ay, Az, and At for the discretization steps of the secondary grid, even if in the FDTD method these quantities coincide numerically with the primary ones. The variables used within the method are the x, y, and z components of the electric and magnetic field E and H, which are attached to the midpoint of the grid edges, and the local values of the material properties at the same locations. The nodes and the associated quantities are individuated by integral or half-integral indexes; (k + 89 for example, E~i,j+(1/Z),k +(1/2)),n+(1/2) stands for EX(iAx, ( j + 89 (n + 89 (Figure 44). With this symbolism, the FDTD time-stepping formulas are, for the time stepping of H x, AYAzP(i+(1/2),j,k)(H~i+(1/2),j,k),n+ AyAt(EY(i+(1/2),j,k
1 -- H~+(1/2),j,k),n)
=
+(1/2)),n+(1/2) - Efi+(1/2),j,k_(1/2)),n+(1/2)
AzAt(EZ(i+(1/2),j+(1/2),k),n+(1/2)-
) --
E~i+(1/2),j_(1/2),k),n+(1/2)
)
(220)
and, for the time-stepping of E x, A yAz~,(i,j+(1/Z),k +(a/z)(E(Xi,j+(1/Z),k +(1/2)),n+(1/2) -- E(Xi,j+(1/Z),k +(1/2)),n-(1/2))
--=
- AyAt(H~i,j+(1/Z),k + 1),, - Hfi,j+(1/Z),k),n) ~~ + AzAt(H~ti,j+ a,k + (a/2)),,
_
H
(i,j,k + (a/2)),,)
z
(221)
Analogous relations hold for the time-stepping of the other components of E and H [Kunz and Luebbers, 1993; Taflove, 1995]. Note that the method seemingly does not make use of global physical quantities, resorting instead to nodal values of local vector quantities. We can, however, observe that each of the field components appearing in these formulas can be considered the ratio of the global quantity associated with a cell and the extension of the cell itself. Interpreting the local field components in this way, that is, as averaged field components with respect to space-like and space-time 2-cells, and remembering that from the local constitutive equations follows
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CLAUDIO MATTIUSSI
I-I y
i,j+ 1/2,k+ 1
HX
i-1/2,j,k+l
i+ 1/2,j+ 1/2,k+
HZ
E y
i,j,k+ 1/2
i+ 1/2,j+ 1,k+ 1
z
/~ ~EXi,j+ 1/2 1/2,k+
X FIGURE 44. The F D T D method makes use of two dual-orthogonal grids staggered by a half step in space and in time. The variables appearing in the time-stepping formulas are the field components of E and H tangent to the grids edges and evaluated in the midpoint of each edge.
x
that B~i+(1/2).j.k).n--ll(i+(1/2).j.k)H~i+(1/2).j.k).n and O(i,j+(1/2),k+(1/2)),n+(1/2 ~(i,j+(1/2),k+(1 /2)E~i,j +(1/2),k+(1/2)),n+(1/2), we can write
AyAzll(i+tl/Z).j.k)H~i+(1/z).j.k).n+
1 =
r
AzAtE~i+(1/2),j+(1/2),k),n+(1/2)
=
(222)
1
AyAtEfi+(1/2).j.k+(1/2)).n+(1/2) = ~ey (i + ( 1/2),j,
k _+ ( 1 / 2 ) ) , n + ( 1 / 2 )
~ ez (i+(1/2),j+(1/2),k),n
) --
+(1/2)
(223) (224)
and AyAzc,(i,j+(1/2).k+(1/2)E~i.j+(1/2),k+(1/2)).n+(1/2
) -- @(dF.j+(1/2),k+(1/2)),n+(1/2)
yAtHfi.j+(1/Z).k+ 1).n = o(hy.j+(1/Z).k+ 1).n '~' ~ AzAtH~i.j+
1.k+(1/2)).,, =
hz ~(i.j+
a.k+(1/2)).,,
(225) (226) (227)
With these definitions the F D T D method can be described as working in terms of global quantities. In particular, the F D T D time-stepping formula
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(220) for H x becomes the following time-stepping formula for (t~bx r
-
-
r
-~- r
--r
~:+(1/2),j+(1/2),k),n+(1/2)
(228)
"Jr- r
and the FDTD time-stepping formula (238) for E x, becomes
O( :j +(1/2),k+(i/2),n+(1/2)
- o(dx,j+(1/2),k+(1/2)),n-(1/2) - o(hty,j+(I/2),k+
1),n
-Jr" o(hx,j+(1/2),k),n Jr- o(h=,j+ 1,k+(1/2)),n -- ~(h=,j,k+(1/2)),n
(229) Comparing Equations (236) and (240) with the time-stepping formulas (188) and (190) of the reference discretization method, we recognize that the former are a particular case of the latter. The signs of the ~be and ~h terms in the right side of Equations (236) and (240) correspond to the incidence numbers appearing in the reference formulas. From Equation (222) we see that the following relation holds true 1
1
H~+(1/z),j,k),,, #(g+(1/2),j,k)AyAz ~b~r+(1/z),j,k),,
(230)
On the other hand, from the equation corresponding to Equation (226) for the case of the x component, we determine 1
H~+Cl/2),j,k),,, = ~--~-~ ~'~'#+(i/2),j,k),,,
(231)
Comparing Equation (230) and Equation (231), we obtain
AyAz
~+(1/2),j,k),n = ]A(i+(1/2),j,k) ~
This is the constitutive equation linking t~ bx applied to Equations (225) and (223), gives .L dx Url(i,j+(1/2),k+(1/2),n+(1/2) -- ~(i,j+(1/2),k+(1/2)
AxAt
O~:+(1/2),j,k),n
to
(232)
0 hx. The same procedure,
~)(eix,j+(1/2),k +(1/2),n+(1/2)
(233)
Equations (232) and (233) are clearly constitutive equations of the simple type [Equation (205)], obtained by a simple extension of the local constitutive equations B = #H and D = eE, exploiting the planarity, regularity, and orthogonality of cells in the meshes adopted by the FDTD method.
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This is even more clear writing Equations (232) and (233) as follows r
(1/2 ),j,k),n
~l~f+(I/2),j,k),n "~X "~{
-- [Ll( i + (1/ 2 ) ' j' k )
AyAz
o(dix,j + (1/2),k + ( 1 / 2 ) ) , n + (1/2) A~ym"~ = e(i,j+(1/2),k + ( 1 / 2 )
(234)
ex
C (i,j + (1/2),k + (1/2)),n + (1/2)
(235)
AxAt
Therefore, we can affirm that the F D T D method implicitly uses the discretization strategy 1 for the discretization of constitutive relations. Substituting these discrete constitutive equations, or their inverse, in the time-stepping formulas (236) and (240), we obtain time-stepping formulas in terms of two global variables only. In particular, the formulas in terms of cb and Od are r
1 --- r
n
AxAt (
1
+ ~YA'Z
e(i+(1/2).j.k+(1/2)
O~c+ ( 1/2),j, k + ( 1/2))on + ( 1/2)
O(~c+ ( 1/2),j,k - ( 1/2)),n + ( 1/2) C'(i + ( 1/2),j,k - ( 1/2))
C,(i + ( 1/2),j + ( 1/2),k)
+
O~c+ (1/2),j + (1/2).k),n + (1/2)
(236)
O~/x+ ( 1 / 2 ) , j - ( 1/2),k),n + ( 1 / 2 ) )
~(i + ( 1/2),j - ( 1/2),k)
and ~l (dxj
+(1/2),k+(1/2)),n+(1/2) AxA't AyAz +
dx
' - O(i,j+(1/2),k+(1/2)),n-(1/2) 1
[,s
1
1
~(bxj+(1/2),k+ 1),n +
~(bxj+(1/2),k),n
~(i,j+(1/2),k)
1)
4)(]xj + i.k + ( 1 / 2 , . ,
#(i,j+ 1 , k + ( 1 / 2 ) )
1
-
~(i,j,k +(1/2))
bx
4),.j.k + (~/2,.,
~] /
(237)
Analogous formulas can be written for the other components. It is interesting to consider also the case of lossy materials, in which case Equation (238) becomes [Taflove, 1995] AYAZ~",(i,j+(1/2),k+(1/2)( E x(i,j+(1/2),k+(1/2)),n+(1/2)
_ E x(i,j+(1/2),k+(1/2)),n-(1/2))
y Y - AyAt(H(i.j+(I/Z).k+ ~ ) . , - H(i.~+(~/2).k).,) "9
+ A z A t ( H ri,j+ 1,k+(1/2)),n
__
g
g
(i,j,k+(1/2)),n)
"-~ A - Y ' ~ ' ~ ( 7 ( i , j + ( 1 / 2 ) , k + ( 1 / 2 )
---
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• ( E~ij+(ll2)'k+(ll2))'n+(ll2)-+Ex(i,j +(1/2),k +(1/2)),,-(1/2)) ,
(238)
2 The new term with respect to Equation (238) represents the charge flowing through a surface (~'y~z) during a time interval ~t, and can, therefore, be written as Ex
x
-~-f "~Z'~(i,j+(1/2),k +(1/2) ) . (i,j+(1/Z)),k +(1/Z)),n+(1/2) q- E(i,j+(1/2),k +(1/Z)),n-(1/2)
)
2
= Q(88 +(1/2),k+(1/2)),n
(239)
Thus, with the new term the lossless time-stepping formula (240) becomes
f,,xj+(l12),k+(l12)),n+(l12)
= Ill(cli~,j+(ll2),k+(ll2)),n-(ll2)
![l(~j+(ll2),k+
1),n
hy hz hz jx -+- @(,,j+(ll2),k),n + @(i,j+ 1,k+(iI2)),n -- @(i,j,k+(ll2)),n -Jr-Q(i,j+(ll2),k+(ll2)),n
(240) and we have the new discrete constitutive equation
Q(J~,j+(1/Z),k+(1/2)),n =
aTa7 AxAt
tT(i'j+(1/2)'k+(1/2))
X(r162
2 (241)
which can be written also as
Q(J~,j+(1/2),k + (1/2)),n
= O'(i,j+(1/2),k+(1/2) X
j+(1/2),k+(1/2)),n+(1/2) -~- r
ex
)
2 AxAt
(242) Note that, contrary to Equations (232) and (233), which are one-to-one discrete constitutive equations, Equation (241) links Qjx to two distinct values of q~ex, corresponding to two consecutive primary time-intervals. This can be explained considering that the space-time volume to which Qjx is associated spans a secondary time interval ~ , which is only half covered by a primary time interval At. From this, the desirability to involve at least two instances of 4~ex in the determination of a single instance of Qjx. To actually obtain from Equation (240) a time-stepping formula, the last term must be
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expressed in terms of electric fluxes only. To this end, we invert the discrete constitutive Equation (233), obtaining x (1/2),k + (1/2)),n + (1/2) C e(i,j+
m
1
AxAt
--
(243) so that Qjx in the time-stepping formula (240) can be expressed as Jx Q(i,j + (1/2),k
+ (1/2)),n
m -
-
A.---{~(~,j+(_~_/Z),k+(X/2)) . (~,j+(1/Z),k+(1/2)),,,+(1/2) + O(~,j+~l/2),k+(1/2)),n-(1/2) e ( i , j + (1/2),k + (1/2) 2 (244) Note how this relation involves two constitutive links, one going from primary to secondary quantities, and one going in the opposite direction (Figure 37). In summary, with the interpretation of physical quantities suggested in Equations (222)-(224), (225)-(227), and (239), the F D T D method appears to adopt a discretization strategy fully consistent with the prescriptions of the analysis of the structure of physical field theories, both in the discretization of the geometry and in the association of global physical quantities to space-time geometric objects. The F D T D method thus appears as a particular instance of the reference discretization method presented above. The time-marching is performed by means of the truly topological time-stepping formulas (236) and (240), supplemented by the discrete constitutive Equations (232), (233), and (241). Note that to determine the discrete constitutive equations, the constitutive relations are implicitly subjected to the simplest of the three kinds of discretization strategies considered above. This leads one to expect that, if this consequence of the original, intuitive approach, which led to the F D T D method, is not properly recognized beforehand, the efforts trying to extend the method to higher orders in space and time, or to nonorthogonal and unstructured meshes, are bound to produce mediocre results or meet with severe difficulties.
2. The Support Operator Method The Support Operator Method (SOM) [Shashkov and Steinberg, 1995; Shashkov, 1996; Hyman and Shashkov, 1997; Hyman and Shashkov, 1999] is an FD technique that permits the derivation of discrete approximations to differential operators, which preserve some properties of the original continuous mathematical model within which the operators to be approximated appear. In particular, the focus is on the simultaneous preservation
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119
of some integral identity that is used in writing a topological law in continuous terms, and of some adjointness relation between pairs of topological statements that face one another in the factorization diagram of the corresponding physical theory. Given this emphasis on integral relations, it is instructive to compare this approach with that of the reference discretization strategy. The discretization of geometry adopted by the SOM is typical of FD methods in that a sufficiently well-behaved set of nodes is considered within the domain in space. For example, in two dimensions it is assumed that by properly joining these nodes one can construct at least a logically rectangular grid, that is one that is homeomorphic to an actual rectangular grid [Shashkov, 1996]. This implies that the resulting grid is, in fact, a cellcomplex, since it derives from the topological distortion of a subdivision of a domain in simple rectangular cells. Therefore, in addition to the 0-cells constituted by the original set of nodes, sets of p-cells with p up to the dimension of the domain are implicitly defined. This constitutes the primary mesh. The SOM does not make explicit use of a secondary mesh. The variables used by the method are the field components perpendicular or tangential to the cells, and associated with the centers of the cells (which, of course, in the case of the nodes are the nodes themselves). As in the case of the F D T D method, this opens the door to their interpretation as representatives of global vector quantities. In fact, apart from nodal quantities, these components appear in the method's formulas multiplied by the extension of the corresponding cell. Therefore, if we think of these variables as averaged values of the field over the cell, their products corresponds to global quantities. Up to this point, we have merely described some quite general premises to FD discretization. It is in the discretization of the field equations that the SOM differentiates itself from a classical FD approach. Instead of discretizing separately the differential operators appearing in the field equations, the SOM selects one of the differential operators to play a privileged role in the discretization. This is called the prime operator. The prime operator is discretized with an FD approach, that is, by substituting its derivatives with finite difference approximations. However, this is done trying to preserve as much as possible the integral properties of the prime operator. For example, if the prime operator is a divergence, a discrete version of Gauss's divergence theorem is required to apply to the discretized operator, which in this case is designed as DIV. The other differential operators that appear in the field problem are then considered for discretization. These are related to the prime operator by some integral relation. We know from our previous analysis that these amount substantially to the continuous counterparts of two properties of
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the coboundary operator: the fact that 66 = 0 (from which the properties such as = 0, curl grad = 0, and div curl = 0 are derived), and the adjointness (with respect to a suitable bilinear form, which puts in duality the corresponding cochains spaces) of the coboundary acting from ordinary p-cochain to produce ordinary (p + 1)-cochains on the primary mesh, with the coboundary acting on twisted ( n - (p + 1))-cochains to give twisted (n - p)-cochains on the dual secondary mesh. For example, if the primary operator is a divergence, the corresponding integral relation is
dd
fv
~odiv A +
fv
A. g r a d ( p = for q~(A. n)
(245)
In this case, to discretize the gradient operator, the SOM puts in duality the discrete spaces of the variables by means of a suitable inner product, and enforces a discrete counterpart of this relation. The resulting discrete operator is marked in some way, to signify its being a derived operator instead of a prime operator. For example, if the divergence is adopted as prime operator, and Equation (245) is used to obtain a discrete counterpart of the gradient, the resulting discrete operator is denoted with GRAD. We have not yet spoken of the constitutive links. The SO M does not adopt a separate discretization of these terms, but includes instead this task in the discretization of some differential operator appearing in the field equations. Therefore, in place of a separate discrete constitutive operator, the SOM produces a discretized compound operator. For example, the operators ~curl, def 1 1 and div ~ def = 7curl5 - d i v e are defined, and they are discretized with the same procedure adopted for the purely differential operators. The only difference is that a bilinear form, which includes the constitutive links, is used to put in duality the spaces of discrete variables prior to discretization. Hence, three types of discrete operators result: a first class of operators determined from prime differential operators, denoted with GRAD, CURL, DIV; a further class of operators determined as derived discrete operators, denoted with GRAD, CURL, DIV; a third class of operators determined as derived operators from compound differential operators, denoted, for example, with DIV ~ and ~CURL,. Finally, the derivatives with respect to time that remain in the semidiscretized model when the differential operators have been substituted by their discrete counterparts are discretized with the traditional approaches, that is, approximating the time derivative with a finite difference formula. In particular the standard leapfrog method is suggested for this task. Examining the workings of the SOM, it is apparent that the method recognizes the necessity to take into consideration a number of structural properties of the field problem. However, this is done when the problem
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121
itself has already been modeled in continuous terms (i.e., the branch on the right has been selected in Figure 1. Therefore, properties such as quantity conservation and the adjointness of operators, which can be easily expressed in discrete terms adopting the discrete mathematical model of the reference strategy, are instead considered first at the continuous level, and only subsequently enforced in a discrete setting. For this reason, the SOM appears to take a long detour to enforce properties that are automatically satisfied using the approach based on the structure of physical theories. Moreover, the unique discrete operator for the representation of topological laws constituted by the coboundary operator, and the related topological time-stepping process, are not considered. In this sense, the SOM is representative of the task thus constituted and the possible pitfalls implied by the search for a structurally sound discretization strategy, which takes as its starting point the continuous mathematical model.
3. Beyond FDTD The analysis conducted above shows that the discretization strategy adopted by the F D T D method is a physically sound one. Moreover, the method is easy to understand and to implement, at least for simple materials. All this makes F D T D a very successful method. This success has logically lead to many efforts focused on the removal of its limitations. These lie mainly in the scarce flexibility of its orthogonal grids in modeling complex geometries, and in the low order of accuracy of the method, both in space and in time. Consequently F D T D extensions to nonorthogonal or unstructured meshes, and to formulas having higher accuracy in space and time, have been presented [Taflove, 1998]. Given the analysis presented in this work, and equipped with the conceptual tool represented by the reference discretization strategy, we know in what direction to look for an extension of F D T D capable of preserving its favorable qualities. The rationale can be stated as follows: "keep two dual space-time cell-complexes
as meshes, keep the topological time-stepping, and improve the discretization of the constitutive relations." Unfortunately, the classical FD approach says instead: "use an expression of differential operators in generic curvilinear coordinates and increase the accuracy of the approximation of derivatives appearing in the partial differential equation." However, this does not lend itself to an easy generalization beyond logically rectangular grids. Moreover, since the derivatives appear in the equations as a consequence of the local representation of the topological operators, a brute-force approach to the increase of the accuracy of their approximation, be they expressed in cartesian coordinates or in curvilinear coordinates, is likely to lead to time-stepping formulas that cannot be considered as derived from a coboun-
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dary operator. We will, therefore, neglect the analysis of the extensions of F D T D based on local viewpoints, such as the classical F D approach, or strategies based on ideas borrowed from differential geometry, which make use, for example, of covariant and contravariant local basis vectors to express the differential operators on nonorthogonal grids [Taflove, 1998]. We will look instead directly to methods that preserve the focus on the discrete nature of topological laws--namely, Finite Volume methods.
C. Finite Volume Methods Broadly speaking, we can say that a numerical method is a Finite Volume method (FV) if, to discretize the field equations of a problem, it subdivides the problem domain in cells and writes the field equations in integral form on these cells. If we accept this definition, we see at once that the FV approach is very similar to that advocated by the reference method. However, the adoption of an integral approach alone does not assure that all the requirements of the physical approach to the discretization are recognized and implemented. For example, one could write an integral statement where topological and constitutive links are mixed, thus missing a fundamental distinction. Moreover, usually the discretization produced by the integral statements of FV does not include the time variable, which is instead subjected to a separate discretization. This opens the door to the possibility of time-stepping formulas that cannot be derived in a natural way from a space-time coboundary operator. From this point of view, the two FV methods for time-dependent electromagnetic problems that we will examine here will turn out to be quite well-behaved, in that they can be both interpreted as particular cases of the reference discretization strategy.
1. The Discrete Surface Integral Method The Discrete Surface Integral (DSI) method was suggested by Madsen [1995] for the solution of time-dependent electromagnetic problems on domains discretized using unstructured grids. The method was first presented for the case of lossless, linear, isotropic materials, but can be easily extended to lossy materials [-Taflove, 1998] and to more complex electric and magnetic constitutive relations. For the discretization of the domain in space the method requires two dual meshes constructed exactly like those of the reference strategy (Figure 33), but for the fact that within the DSI method there is no mention of the distinction between external and internal orientation. To simplify things with respect to a generic cell-complex, we will assume that all 1-cells are straight lines and all 2-cells are planar. The variables used as unknowns by
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FOR PHYSICAL FIELD PROBLEMS
123
the method are at first sight not global ones but are instead field quantities associated with the edges or the faces of the two grids. However, we proceed to show that in this case also the field quantities are used in such a way that global quantities are actually intended. Let us start with the quantity associated with the primary 1-cells z]. This quantity is defined by the DSI method to be the projection E . s k of the electric field intensity vector E - - a s s u m e d to be constant along the c e l l - onto the primary 1-cell r] represented as a vector sk. It is apparent that this means that a global value is actually considered associated with each primary 1-cell. In fact, if E is assumed constant also during a time interval At, we can consider directly the global space-time quantity qSf,= E.skAt thought of as associated with a space-time primary 2-cells, since this is the form in which E always appears in the DSI formulas. Correspondingly, the quantity associated with the secondary 1-cells ~] is the projection H. ?,k of the magnetic field intensity vector H - - a s s u m e d to be constant along the c e l l - - o n t o the secondary 1-cells ~] represented as a vector gk. This association can be also extended to consider the space-time global quantity =
H.
With each primary 2-cell is associated a full magnetic flux density vector B, and with each secondary 2-cell is associated a full electric flux density vector D, both assumed to be constant over the corresponding cell. Even if these quantities are given as full vector ones, they appear in the DSI discretized Maxwell's equations only as B.N~ and D.N~, where N~ and N~ is the so-called area-normal vector, defined so that the two scalar products are actually the magnetic flux ~b} = B. Ng and the electric flux ~,~ - D - l ~ , associated with the corresponding cells. In this way we have interpreted all DSI variables as global quantities. We can now consider the DSI discretization of Faraday's law and Maxwell-Amp6re's law in the light of this interpretation of the variables. Faraday's law at time tn+(1/2 ) = (n -+- 1) At is written for a primary 2-cell v~2, represented by the vector N i, as dB,,+(1/2). Ni =
_
dt
_
J a~
E,+(1/2)" dl
(246)
To discretize the time derivative the DSI method adopts a time-centered leapfrog algorithm, which sets
dBn+(1/2 ) dt
B.+
1 -
At
B,_ 1
(247)
Substituting Equation (247) in Equation (246) we obtain the DSI time-
124
CLAUDIO
MATTIUSSI
stepping formula for magnetic flux B.+
I"Ni
=
B,-N i
--
At l da
En+(1/2 ) "dl
(248)
Remembering the expression of the boundary in terms of incidence numbers, Equation (248) becomes B.+ 1" N, = B.. N~ __ ~ El:~,~?[3E,+(1/2)" skAt
(249)
k
(with the uncertainty on the sign of the last term due to the usual dilemma regarding the relative default orientation of primary 1-cells with respect to the default positive direction of E). With the interpretation of the variables as global quantities given above, Equation (249) becomes (~ib,n +1
"-
~)i,n b -Jr- E ET'i2' T'kl]~)k,e n+ (1/2) k
(250)
which is exactly the topological time-stepping formula (188) of the reference discretization method. The same procedure can be applied to show that the DSI time-stepping formula for D is Dn+(1/2)" Ni = Dn-(1/2)" Ni + At
H,. dl
(251)
which reduces to the reference topological time-stepping formula (190). It remains now to examine how the DSI method proceeds to the discretization of the constitutive relations. We assume as given the local constitutive equation B =/~H, and we consider how it is used to determine a discrete link going from 4)b to Oh. The DSI method adopts a reconstruction-projection method that is a particular case of the discretization strategy 2 described above for the discretization of the constitutive relations, which uses a reconstruction-projection process. The reconstruction is performed with the following procedure. Consider two adjacent primary 3-cells, ~:~ and ~:2 (Figure 45). They have in common a primary 2-cell ~:~. The boundary of ~:~ is composed by primary 1-cells whose boundaries constitute a collection of 0-cells r~'. With each of these 0-cells the DSI method associates first two magnetic flux density vectors Bl,m and Bz,m, one for each of the two 3-cells, ~:~ and ~:2. Each of these vectors is derived from a system of equations asking the fluxes calculated integrating them on three 2-cells (over which they are assumed as constant) to equal the fluxes associated with these same cells as DSI variables. In more detail, calling r 89 r~ the other cells (besides the common cell ~:~) belonging to the boundary of ~:~, which meet in the node ~', and calling N c, Nj, N k the corresponding area-normal vectors, and calling ~b), 4)b, 4~ the
N U M E R I C A L M E T H O D S FOR PHYSICAL FIELD PROBLEMS
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FIGURE 45. The Discrete Surface Integral method adopts a reconstruction-projection strategy for the discretization of the constitutive equations, which is based on the boundary cells of pairs of adjacent 3-cells.
variables associated with these 2-cells, the DSI method sets = BI,
Nc
~bbl= BI,,,, N j
(252)
= BI,,, Nk and determines Ba,,, in terms of 4~c b, 4~, and 4?. The same process is repeated for the adjacent 3-cell z 2 to determine B2,m, and for all the nodes z~ belonging to the common 2-cell z~. The information constituted by all the BI,,, and B2, m thus determined is then merged using a weighting formula, to produce finally a single vector B. The seminal paper on DSI [Madsen, 1995] examines three weighting formulas for this task. The vector B is then assumed to be the reconstructed, constant field within the two adjacent 3-cells, z~ and z 2. It depends on the values of the variable 4~c b associated with the common 2-cell r~ and on the values 4)b associated with all the 2-cells belonging to the boundary of z3x and r2, which 1 touch z~. The inverse local constitutive equation H = 5B is then applied to the reconstructed field. The resulting field H must at this point be projected on the secondary 1-cell ~], dual to r~, to determine H ' g c. This is done using
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C L A U D I O MATTIUSSI
the. formula
B" sc
H'gc = ~ _ /~c
(253)
where ftc is obtained averaging ~ along ~. Finally the global space-time value Och = H- gc~'is determined multiplying the result of Equation (253) by the time step ~ . This whole process produces a discrete constitutive link, which relates Och to the values q5b on which the reconstructed field B depends. A similar procedure can be applied to discretize the constitutive equation D = eE, obtaining a relation linking the global space-time quantity q5e associated with each primary space-time 2-cell to the values O~ associated with a small number of secondary neighboring 2-cells. Note that for boundary 2-cells the reconstruction procedure described above is based on a single 3-cell. In summary, this analysis shows that the DSI method is, like the FDTD method, fully compliant with the prescriptions of the reference discretization strategy. It adopts a topological time-stepping formula for the global electromagnetic variables, although this remains hidden due to the use of local field quantities in the original description of the method. Compared to FDTD, the DSI method defines in more general terms the quantities involved, so that its time-stepping formulas apply to generic unstructured grids (only provided they are two dual cell-complexes) on which they preserve their topological nature. Moreover, the DSI method adopts a more sophisticated approach than FDTD to the discretization of constitutive equations, since the DSI approach is based on a more complex reconstruction-projection strategy. All these properties make of the DSI method a generalization of FDTD, which, from the point of view of the structure of physical field theories, preserves the favorable characteristics of that method. Considering in detail from the same point of view its reconstructionprojection strategy, the DSI method appears, however, to be far from optimal. The reconstruction strategy is actually focused on the determination of nodal field quantities and does not make use of edge elements. It is, therefore, likely that experiments with different reconstruction-projection operators more intimately related to the cochain concept would lead to further improvements of the method, not only in terms of compliance with the structure of electromagnetism but in terms of accuracy also.
2. The Finite Inteyration Theory The Finite Integration Theory (FIT) method is a Finite Volume method for time-dependent electromagnetic problems, which was developed independently of FDTD [Weiland, 1984; Weiland, 1996]. It is interesting to examine
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127
this method for two reasons. First, it underwent a series of improvements from the time of its first appearance in the literature, which are making it more and more similar to the reference discretization strategy described above. Second, it distinguishes well the various phases of the discretization process for a field problem. In a first phase of its development [Weiland, 1984] the discretization of geometry adopted by the FIT method was based on two dual orthogonal grids, G and G. The idea of two kinds of orientation was not explicitly mentioned, but its consequences in terms of association of physical quantities to the two grids were implicitly used. In a more recent reformulation of FIT [Weiland, 1996], the orthogonal grids are abandoned and the fact that the two meshes need only be cell-complexes is recognized. The new kind of mesh G is constructed by partitioning the spatial domain into volumes V i, whose nonempty pairwise intersections is a set of surfaces A J, whose pairwise intersection is in turn a set of lines L k, whose pairwise intersection is in the end a set of points P~. Comparing this procedure with the definition of a cell-complex given above, we recognize in the resulting structure G, our primary cell-complex K, and in the sets {Vi}, {AJ}, {Lk}, and {pl}, four sets of p-cells {z~}, p = 0 , . . . , 3. The dual mesh (~, which corresponds to the secondary cell-complex K, is constructed defining for each V i of G a dual point /sz located within V i, and proceeding then to define the other dual objects Lj, ~k, and pl. The discretization of fields, like that of the geometry, changed with the development of the method. Originally [Weiland, 1984] the quantities considered were the field components tangent to the lines of the grid in the case of field intensities E and H, and the field components perpendicular to the cells in the case of flux densities B and D. These components were assumed as evaluated at midcell, and were subjected to numerical integration to obtain an approximation of the global quantities appearing in Maxwell's equations in integral form. More recent formulations of the FIT [Weiland, 1996] show that it has been recognized in the meantime that in writing these equations there is no reason to introduce the local field variables first. Consequently, the variables considered became the global field quantities associated with the geometric objects of the meshes. This process, however, was not fully carried out to include space-time geometric objects. Therefore only global quantities associated with space-like objects are considered. These, adapting the notation to that used in the present work, are the electric voltages Vf, associated with the primary 1-cells, z] = Lk; the magnetic fluxes qS~ associated with the primary 2-cells, z~ - AJ; the magnetic voltages F~ associated with the secondary 1-cells, ~{ - U; the electric fluxes ~ and the electric currents I/s associated with the secondary 2-cells, ~k2- j k; and the electric charges Qf associated with the secondary
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3-cells, ~ = ~t. In formulas t" V~,= [ E J,
(254)
~b~ = f, B {
(255)
F h = ~~ H
(256)
r i b
q/~ = {- D 3~
(257)
I~ = [ J (258) J~ /. (259) Qf = | P J~ Note that considering only the geometric objects in space, two distinct quantities of the same physical theory appear as associated with the same geometric object ~'~. The variables just defined are grouped by FIT into vectors V e, (I)b, ~h, ~a, Tj, and Q", which we can interpret as natural representations of cochains. The discretization of topological laws at this point follows easily. Maxwell's equations in integral form with respect to space, but in differential form with respect to time, are considered [Equations (23), (28), (29), and (24)]. Using implicitly the coboundary operator in space, these equations are semidiscretized, that is, the space-like part of the topological relation is written in terms of the above cochains, leaving the time derivative in its original differential form. In matricial form, this reads d(D b
d--t + D2.~V~ = 0
(260)
D3,2(I )b -- 0
(261)
- ~h = [j + D2,1
(262)
dq,~ --~
dt
6 3 , 2 t I -/d -'- (~P
(263)
where_ D2,1 and D3, 2 are the incidence matrices on the primary cell-complex, and D2,1 and D3, 2 those on the secondary complex. The following relations hold true (Wieland, 1996): D2.1 = I)T2.~
(264)
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
129
D 3 , 2 D 2 , 1 -- 0
(265)
D3,202,1 = 0
(266)
Since the incidence matrices are the matricial representations of the coboundary operator, these relations correspond to the abovementioned adjointness of pairs of coboundary operators acting on the primary and secondary meshes, and to the relation 66 = 0 considered on the primary and secondary mesh. The constitutive equations are then discretized. The literature on the method is somewhat vague about the details of this process. The method used seems to correspond to Strategy 1, above, that is, two dual cells are considered and the local constitutive equation is extended to the global quantities associated with these cells. For example, to9 discretize the constitutive equation B - / z H , two dual orthogonal cells, r~ and ~], are considered, and the corresponding global quantities are assumed as linked by a relation of the kind ~bb. J F~ = Cu'j
(267)
The coefficients C,, j constitute a matrix C u, which is diagonal for simple constitutive equations and meshes having orthogonal dual cells, but can be nondiagonal in more complex cases The discrete constitutive equations are, therefore, linear relations of the kind ~/d __ CeV e
(268)
(I)b -- C # F h
(269)
I~ = C~V e
(270)
where the subscript in the term i~ signals that there can be other contributions to the electric current, besides this one. Using these equations, the semidiscretized equations (260) through (263) can be rewritten in terms of two cochains only, which the FIT method chooses to be (I)b a n d V e. In particular, besides Equation (260), which already depends on these two quantities only, the time-dependent Equation (262), which corresponds to Maxwell-Amp6re's law, becomes ---(CeV
dt
e) --F-D 2 , 1 C 2 l(I)b = IJ
(271)
The set of semidiscrete equations obtained in this way are called Maxwell Grid Equations.
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Note that up to this point, no time-stepping procedure has been defined. This is necessarily based on the two time-dependent semidiscretized equations (260) and (271). Various strategies for the discretization of the time derivatives remaining in these equations are considered. In particular the usual leapfrog method is pointed out as the method of choice in most cases. For a time step At this gives the following time-stepping formulas [Weiland, 1996] O~ +1 = ~b,b - AtD 2 1Vne+(1/2)
(272)
Vne+(1/2) = Vne_(1/2) -1t- A t C ~ l ( O 2 , 1 C ; 1Onb -- l J )
(273)
These formulas cannot be considered topological time-stepping relations. To arrive to a topological time-stepping relation they must be first rearranged as follows ob + 1 = Onb -- D 2 , 1 AtV e +(1/2) CeVne+(1/2) = CeVne_(1/2) -1t- ( O 2 , 1 A t C ;
(274) l o b -- A t i j)
(275)
and then, considering the constitutive relations (268) and (269), introducing the cochains qjh, (I)e, and I~j, and putting AtFh= qjh, AtVe= o e and AtiS= Qs they can be rewritten finally as topological time-stepping relations oh + 1 = Onb -- D 2 , 1 O n e+ ( 1 / 2 ) "d "d "h klJn+(1/2 ) -- kI'/n_(1/2 ) -1t- (B2,1klJn --
(276) Q~)
(277)
which correspond to those of the reference discretization strategy. In summary, the FIT method appears to have recognized, in the course of its evolution, the desirability of adopting a number of features that are suggested by the structural analysis of physical theories. These include the choice of cell-complexes to discretize the domain and the priority of global physical quantities associated with geometric objects over local field quantities and their adoption as the method's variables. Moreover, the distinction of topological laws from constitutive relations was built in the method from the start, along with the preservation in the semidiscrete system of equations, of many structural properties of the continuous model of the original problem. The strategy adopted for the discretization of constitutive equations appears, however, quite elementary. Moreover, the method falls short of recognizing the desirability of a truly space-time approach to the discretization. The resulting choice of variables in the time-stepping formulas, and the time-stepping formulas themselves suffer from this oversight. Even with the adoption of the leapfrog time-stepping, the interpretation of the FIT
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131
time-stepping formula as a topological time-stepping appears artificial, while the properties of the continuous mathematical model that were preserved in the semidiscrete model are at risk to be lost in the time discretization step.
D. Finite Element Methods Originally, the Finite Element (FE) method was conceived of as an analytical tool for solid mechanics, and its first formulation was based on a direct physical approach [Burnett, 1987; Fletcher, 1984]. Given its flexibility with respect to F D methods, and the good results produced, the FE approach was applied to many other fields, with the variations required by the nature of the new problems. A whole class of FE methods ensued, which were soon given a rigorous mathematical foundation using the ideas of functional analysis. In spite of this later formalization, the origins of the method leads one to expect that a certain similarity exists between the FE approach to discretization and the "physical" one of the reference discretization strategy presented above. Let us, therefore, examine the FE methods from this point of view. We must first define what we intend to consider as an FE method, and this we will do in operative terms. To speak in concrete terms, let us consider a simple electrostatic problem. We assume that a distribution of charge p is given in a domain D, along with suitable boundary conditions along c~D, and we seek the electrostatic potential Von D. We know that the field equations for this problem can be factorized into the following pair of topological equations -grad V = E
(278)
div D = p
(279)
supplemented by a constitutive equation of the kind D = f~(E)
(280)
The FE discretization procedure starts with the subdivision of the n-dimensional domain D in elements. In the simplest cases the elements correspond to the n-cells of the reference method, and define a mesh in the domain. The field quantities that have been selected as unknowns are then given a discrete representation. This is done in terms of a finite number of variables associated with geometric objects, which belong to the mesh. In our case, since the unknown is a 0-field, these objects are a set of 0-cells, the so-called nodes in FE terminology. Two possibilities are open at this point for the discretization of the equations: the variational approach and the weighted residual approach [Fletcher, 1984]. Given the greater generality of
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the latter, we will consider the weighted residual approach as characteristic of FE methods. To apply this technique, the complete field equation is considered, that is, Equations (278), (279), and (280) are reassembled to give - div(f~(grad V)) = p
(281)
The first step of the FE discretization of this continuous formulation of the problem consists in a reconstruction of the unknown fields based on its discrete representation, using a set of shapefunctions sj(r) (see the section on Edge Elements, above), as follows V(r) = ~ Vjsj(r)
(282)
J
This transforms Equation (281) into -div(f~ (grad ( ~
Vjsj)))=p
(283)
Despite the adoption of a discrete representation for the unknown field, Equation (283) is still a partial differential equation. To obtain from it a system of algebraic equations, a set of weightfunctions wi is selected, and the following set of residual equations is written (284) To obtain a sparse coefficient matrix, FE methods adopt shape and weight functions having local character. Therefore, the support of each weight function is a small subdomain of D, which is in fact a 3-cell that we can denote with r~. Using the notation for weighted integrals introduced above, we can rewrite Equation (284) as follows
The left side of each of these equations can be integrated by parts, giving
-f,~(w,,~)f~(grad(~Vjsj))=fw,~,PVi
(286)
where the meaning of the term on the left side is defined by Equation (172). The set of equations represented by Equation (286) is the system of algebraic equations produced by the FE method.
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We can interpret this strategy in light of the reference method. The field V is first reconstructed starting from the 0-cochain V v = { Vj}, and using the shape functions sj. We can represent this as the action of a reconstruction operator R v as follows V = R~(V ~) Next, the local topological Equation (278) is applied to the reconstructed field, giving the E field. The constitutive Equation (280) is then applied to E, obtaining the electric flux density D. For each weight function, that is, on each spread cell wjz~, the following topological equation is finally imposed
fo
D = f w p Vi
(287)
Compared to the reference strategy, which enforces both topological equations in discrete terms on crisp cells, the approach just described differs in its applying one topological equation in differential terms, and the other in integral terms on a spread cell. The difference could be reduced by reformulating the reconstruction of E and making it start from the ve cochain, and then applying to it the corresponding topological equation in coboundary terms. In any case the projection step is not present for spread cells, which do not require it. Note that if the reconstruction is based on ve, edge elements and not nodal interpolation must be used to obtain a physically sound reconstruction of E. This is always true in the case of magnetostatics problems, since the magnetic potential A is a 1-field and a correct reconstruction must start from the cochain Va and use (ordinary) 1-edge elements. The realization of this fact was one of the reasons that lead to the introduction of edge elements by the computational electromagnetics community. This analysis reveals also the different role of shape functions and weight functions in the discretization process. Shape functions are used to reconstruct the fields in order to approximate the constitutive equations, whereas weight functions define the spread cells which constitute a continuous counterpart of the secondary mesh to which the corresponding topological equations are applied. With different joint choices of the two sets of functions we obtain different categories of methods. If the weight functions wi are the characteristic functions of their support z~, that is, if
wi =
in r3 . outside z~
(288)
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C L A U D I O MATTIUSSI
then the method is called a subdomain method. Considering Equation (287), we recognize that a subdomain method is actually an FV method, since it applies the topological equations to a set of crisp cells z~. If the weight functions coincide with the shape functions, that is, if wi(r) = s,(r), Vi
(289)
then the method is called a Galerkin method. Other choices are, of course, possible and gives rise, for example, to the so-called Collocation method, and to the Least squares method [Fletcher, 1984]. The choice corresponding to the Galerkin method is undoubtedly the most used in FE. This is linked to the fact that, when a variational formulation exists for the problem, Galerkin's choice gives a system of equations that corresponds to that derived from the variational approach. Considering their different role in the discretization process, the systematic choice of coincident weight and shape functions appears, however, to be questionable. If edge elements are adopted as shape functions on the grounds of physical considerations linked to the association of physical quantities with geometric objects, then, in the same spirit, weight functions should be chosen in order to impose in an optimal way the topological equations, and it might turn out that some kinds of shape functions are not ideally suited to this task. For an analysis of the different role of shape and weight functions from a functional analysis viewpoint see Schroeder and Wolff [-1994]. 1. ~me-Domain Finite Element Methods
The FE discretization strategy exemplified above does not lend itself easily to the discretization of time-dependent problems. It has been noted [Fletcher, 1984] that the classical FE method is intrinsically "elliptic," in the sense that it solves problems by "propagating" simultaneously on the whole domain the source and boundary conditions of the problem. Therefore it is ideally suited to the solution of boundary-value problems, but does not apply well to initial-value problems. To adapt the nature of FE to a transient problem defined in a time interval [to, t1], one should consider space-time shape functions s(r, t) and weight functions w(r, t), and transform the initial-boundary-value problem in a boundary-value problem, generating in some way the missing boundary condition at the final time instant t = t l. This can be done, for example, by putting t l = oo and using the steady-state solution of the problem with time-infinite elements, or using finite elements and a t l great enough to make the solution at that time sufficiently similar to the steady-state condition [Burnett, 1987]. Understandably, neither of these approaches enjoyed great popularity.
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
135
A first alternative to this approach is the adoption of the separation of variables technique. Assume that, as usual, the problem constituted by Faraday's law (8) and Maxwell-Ampbre's law (13), with the simple constitutive equations (20), (21), and (22) and with suitable initial and boundary conditions, is to be considered. As for the electrostatic problem above, we first combine the equations into a single partial differential equation, for example [Lee et al., 1997] curl
(~
\ 02E ~E 0J i curl E ) + e + a + = 0
/
-Y
-bY
(290)
where Ji are the impressed currents. The space-time shape functions are expressed as products of functions that depend separately on space and time, and the unknown field is reconstructed as follows E(r, t) - ~ Vf(t)sj(r)
(291)
Here the shape functions are assumed to be 1-edge elements, and the vector of coefficients {be(t)} can be considered a time-dependent cochain ve(t). Next, a set of suitable weight functions wi(r) is considered at a generic time instant t, and the weighted residual method is applied to Equation (290), resulting in a system of ordinary differential equations d2V e
dV e
A dt: + B - - ~ + C V
~+d=0
(292)
where A, B, and, C are matrices and d is a vector. This system of equations is finally discretized and solved using some time-stepping method for ordinary differential equations. The approach just examined, due to the adoption of edge elements for the reconstruction, has in common with the approach of the reference discretization strategy the discrete representation of a field as a cochain. However, the similarities stop there, since the discrete representation of fields does not extend to space-time and this distinction of topological and constitutive equations is not recognized. Moreover, contrary to the case of the electrostatic problem above, where the distinction was not explicitly recognized by the FE approach but could be considered implicitly built into the method, disentanglement is not possible here since Equation (290) mixes inextricably the two time-dependent Maxwell's equations and the constitutive equations. The same holds true for the other approach to time-dependent problems that preserves to the problem the "ellipticity" suited to the classical FE approach, that which considers time-harmonic fields [Jin, 1993]. In that case
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Equation (290) becomes
curl (~ curl E) - coZeE +jacoE +jcoJ, = 0
(293)
where not only are the equations mixed, but the possibility of a space-time approach is also definitely lost. Thus it seems that in spite of its physical origin, the classical FE approach is not capable of producing a truly physical discretization of time-dependent problems. This is even more surprising if one considers that to FE we owe ideas such as that of edge elements, of the reconstruction-projection strategy and of the error-based strategies for constitutive equations discretization. FE practitioners are aware of the problem constituted by the lack of a convincing FE treatment of transient problems, and in recent times have considered with interest the simplicity and effectiveness of the FDTD [Lee et al., 1997]. They have realized in particular that to obtain a physical discretization, one has to start from the factorized equations, that is, consider separately the two time-dependent Maxwell's equations, and the constitutive equations. Let us examine two methods that adopt this discretization philosophy.
2. Time-Domain Edge Element Method An FE-like Time-Domain method directly based on the two time-dependent Maxwell's equations has been suggested, with minor variations, by many authors. This method is described in the survey paper on Time Domain FE methods by Lee et al. [1997]. It adopts a single discretization mesh for the domain in space and defines as variables the global quantities associated with the p-cells of this mesh. Contrary to the reference method, therefore, we do not have two distinct dual meshes. However, since both the primary and the secondary variables are associated with the cells of the unique mesh, we can assume that two "logical" meshes, which have different kinds of orientations and share the same geometrical support, are implicitly defined. ~i for every i and for n = 0, 1 2, 3. With this In other words, we have ~:,i = ~:,, provision the variables of the method are defined like those of the FIT method, by Equations (254)-(258), and are the electric voltages Vke, the magnetic fluxes ~b~, the magnetic voltages F~,, the electric fluxes ~ , and the electric currents I{. Therefore, the fields are correctly represented by cochains, which we denote as V e, (I)b, ~,h, ~d, and -j, I like those of FIT. According to the FE tradition exemplified in the electrostatic example above, the method then proceeds to the reconstruction of the fields E, B, H, D, and J, using as shape functions a set of 1-edge elements s~, and a set of
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METHODS
FOR PHYSICAL FIELD PROBLEMS
137
2-edge elements s 2, as in E = ~ VkeS1 : Re(V e) k
H = Z F~,s/, = Rh(F h) k
D - ~ ~tiaS i2 = Rd(~t jd )
(294)
i
B._
Z ~is~ b 2 = Rb(O b) i
J = ~ J-iJ2_si -- R j ( i j) i
These expressions are substituted into Maxwell's time-dependent equations and then the collocation method is applied to the resulting equations. This means that Maxwell's equations are applied in integral form to the 2-cells. After application of Stokes's theorem, this produces (295) z~
~ i
FkSk +
r
{ =
j 2
Iis i
(296)
which corresponds to
9 / s?=O d s~-~~/~b/afros 2=~/I~f~ S2
(297) (298)
Remembering the characteristic property of edge elements expressed by Equation (219), and expressing the boundaries of cells in terms of incidence numbers, this corresponds to dO b dt
-f- D 2 , 1 v e --- 0
(299)
+ D2,1 ~'h = IJ
(300)
dae d
- ~
dt
where D2,1 is the incidence matrix between 2-cells and 1-cells of the mesh. Note that this corresponds to the semidiscrete equations (260) and (262) of the FIT method. This is not a surprise, since we are actually performing the same steps of the FIT method. The fact that a reconstruction of the field
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C L A U D I O MATTIUSSI
quantities is performed before the enforcement of Maxwell's equations is a heritage of the FE approach, but appears completely superfluous, since the subsequent projection performed while enforcing the equations in integral form reproduces exactly the starting cochain. This is in fact the characteristic property of edge elements, as expressed by Equation (219) or Equation (210). The reconstruction is actually required only in a later phase, that is, when it is time to determine the discrete constitutive equations, expressing V e in terms of qjd, and ~,h in terms of ~b. This is done imposing on the reconstructed fields [Equation (294)] the constitutive equations, and then projecting the result to obtain a cochain, according to Vke = f~ -1 D = p e ( f _, (Rd(tpd))) ~
(301) r~ = f J,
m
1 B
-
-
-
Ph( fi,- l (Rb(Ob )))
This, after substitution and integration, gives the matricial links V e -- C~-, ~ d
(302)
~,h = C~-, 9 b
(303)
Substituting these links in the semidiscrete equations (299) and (300), and discretizing in time using a leapfrog scheme, we have (304)
ob + 1 = Onb -- D2,1 AtC,,-, ' ~ d k- l"J n + ( 1 / 2 )-- q,"n-(1/2) + D2,1 ArC
AtF
(305)
and finally, putting F,:-, = AtC~-,, F,-, = AtC~-,, and 0 J = At]"j, gives ,~ +1 = O , b- D 2 , 1 F,.-, ~d k l J n + ( 1 / 2 )_- q,.n-(1/2) + D 2 , 1 F ~ - ' O h _ O J
(306) (307)
These time-stepping formulas coincide with those of the reference method, Equations (192) and (193). In particular they could be put in the form of Equations (211) and (212), since the discrete constitutive links are determined using a reconstruction-projection strategy. In summary, the Time-Domain Edge Element method described above appears to be in fact a particular FV method, which adopts coincident primary and secondary meshes, applies the topological equations in terms of cochains (the premature reconstruction of fields prior to topological equations application being actually immaterial), and discretizes the constitutive equations using a reconstruction-projection method based on edge
NUMERICAL METHODS FOR PHYSICAL FIELD PROBLEMS
139
elements. Contrary to the reference discretization strategy, the two meshes do not stand in geometric duality, and the global variables are not considered associated to space-time geometric objects. However, thanks to its applying the topological equations to crisp cells, its time-stepping formulas can be considered as implementing a topological time-stepping. 3. T i m e - D o m a i n Error-Based F E M e t h o d
The examples given so far of FE methods, and the accompanying discussion could have created in the reader the impression that it is not possible to build a truly "physical" time-domain method using an FE approach based on spread cells. To prevent the formation of this premature conclusion we present now an example of a interesting error-based time-domain FE method that uses spread cells [Albanese et al., 1994; Albanese and Rubinacci, 1998]. The method is based on the use of potentials both on the primary and on the secondary side. To this end, a slightly modified form of Maxwell's equations must be considered, that is, the set 0B curl E + ~ = 0
(308)
0D T = 0 c~t
(309)
curl H
where D r is a modified electric flux density, which includes the current density term. In this way both these statements appear as flux conservation statements, and the corresponding quantities admit two potentials A and W, such that [Albanese et al., 1994] c~A c~t
E =
H
=
0W c~t
~
(310)
(311)
B = B o + curl A
(312)
D r = Dro + curl W
(313)
The potentials A(r, t) and W(r, t) are formally reconstructed using edge elements as follows A = Ra(dl[ a)
(314)
W = R~(F w)
(315)
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CLAUDIO MATTIUSSI
where ~a and F w are the corresponding cochains. Next, Equations (310)(313) are used to determine the fields. This assures that the fields satisfy Equations (308) and (309). To enforce the constitutive equations, an error density function ~(E, D r, B, H) is defined, following the criteria sketched in the definition of Equation (215), and is integrated in space over the domain D, and in time over a time step At, giving an error functional g =
ft
tn +1 f O ~(E, DT, B, H)
(316)
n
Given Equations (314) and (315), the error functional o~ can be expressed in terms of the cochain ~a and F w. Therefore, a minimization problem based on ~ can be established at each time step, as follows 1, F ; + I }
=
min {ou,"+,,r;'+,} (317)
thus allowing the time-stepping of the potential cochains. From the physical point of view this approach to the solution of Maxwell's equations leaves much to be desired. In particular this is true for its use of a modified form of Maxwell's equations. However, it appears to be a promising idea towards the determination of a physically consistent method based on the traditional approach of FE methods, and for the extension of the error-based methods to the case of time-dependent problems, either adopting the FE or the FV approach.
g. CONCLUSIONS
We have presented a set of conceptual tools for the formulation of physical field problems in discrete terms. These tools allow the representation of the geometry and of the fields in discrete terms, using the concept of oriented cell-complex, and those of chain and cochain. Moreover they allow us to bridge the gap between the continuous and the discrete concept of field by means of the idea of limit systems. The analysis of the structure of physical field theories is based on these tools. This analysis unveils the importance of thinking of physical quantities as associated with space-time oriented geometric objects. It shows also that these objects must be thought of as endowed with one of two kinds of orientation. Moreover, this analysis exposes the distinction of topological laws from constitutive relations showing their different behavior from the point of view of their discretiza-
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141
bility. It clarifies also that a privileged discrete operator--the coboundary operator--exists for the representation of topological laws. A reference discretization strategy, which complies with these concepts, has been presented. It is based on the idea of topological time-stepping for time-dependent equations, which operates on global quantities and derives from the application of the coboundary operator in space-time. It was then shown how topological time-stepping can be combined with different strategies for the discretization of the constitutive relations. In particular, three of these strategies were presented and examined in detail. Analyzing the operation of a number of popular methods, we have shown that there has been a steady tendency of numerical methods devoted to field problems, towards the adoption and inclusion of techniques that adhere to the philosophy described above. In particular we have revealed that many methods can be thought of as implicitly adopting the topological timestepping procedure. According to the signaled trend, even if the concept of topological time-stepping seems to have eluded the creators of these methods so far, we can expect it to be recognized and included explicitly in future formulations of these methods. In the long run, this trend will probably lead to classes of methods, which mix the best features of the various methods. In particular, we have shown that methods such as Finite Differences and Finite Volumes, which do well with regard to time-stepping, usually fail to give the constitutive relations an adequate treatment, thus ending with very crude discretizations that are scarcely applicable to nonstructured grids. On the contrary, Finite Elements methods, which discretize well the constitutive relations and can deal easily with arbitrary meshes, fail with regard to topological laws, especially when applied to time-dependent problems. Thus, the time seems ripe for the combination of the best features of these categories of methods, with the devisement of methods that discretize carefully both topological laws and constitutive relations, bringing to the field of unstructured meshes the advantages of a correct topological time-stepping. In particular, the joint use of error-based discretization strategies for the constitutive relations along with topological time-stepping schemes seems a promising and as yet unexplored field of enquiry. As anticipated in the Introduction, we have not considered here questions such as the rate of convergence, the stability, and the error analyses of the methods. In the light of the present discussion it is, however, worth making at least one observation: the tendency of the various approaches towards the adoption of a number of common ideas includes the use of global variables associated with geometric objects for the discrete representation of fields. Therefore, it also seems logical to focus on the error analyses on the global quantities, instead of applying these analyses to the local quantities
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that are reconstructed from global ones once the numerical problem has been solved. The error, which derives from this last step, is one of cochain-based field function approximation, and is derived from a process of reconstruction of the field functions, which starts from the abovementioned global values. This error is obviously relevant to the solution of physical field problems, but can be considered separately from the previous phases of the numerical method. For example, the final field reconstruction can be conducted with different criteria with respect to possible reconstructions that took place during the discretization phase. Finally, the emphasis placed in this study on a discrete approach to the modeling is not intended to indicate that the alternative continuous approaches should be abandoned in the near future or considered "bad" approaches. These alternative approaches are needed today and will continue to be so in the future, since a discrete approach places some constraint on the relation between the problem to be solved and the resources required to actually do this. There will always be cases where a solution is sought for which the discrete approach presented here appears in a particular moment not to be feasible. Going back to the theme of the introduction, since good numerical mathematics is also the art of the possible, to cover these cases the ingenuity of the mathematician will be required to produce for these problems a formulation that is numerically feasible. Usually this requires the embedding in the discrete formulation of a problem of the physical and mathematical knowledge available regarding the problem and the general behavior of the solution. This includes the exploitation of the properties of the continuous mathematical model. We see in our times examples of this in spectral methods and compact finite difference methods applied to fluid dynamics. However, as time goes by, we can expect that the approach to the discrete formulation of field problems outlined in the present work will be found to be numerically manageable for a steadily widening range of problems, and that for these cases this approach will be recognized and adopted as the method of choice
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Albanese, R., and Rubinacci, G. (1998). Finite Element Methods for the Solution of 3D Eddy Current Problems, In Advances in Imaging and Electron Physics (P. Hawkes, Ed.), 102, 1-86. Baldomir, D., and Hammond, P. (1996). Geometry of Electromagnetic Systems. Oxford: Oxford University Press. Bamberg, P., and Sternberg, S. (1988). A Course in Mathematics for Students of Physics." 1-2. Cambridge: Cambridge University Press. Bateson, G. (1972). The Logical Categories of Learning and Communication. In Steps to an Ecology of Mind. New York: Ballantine Books. Bellman, R. (1968). Some Vistas of Modern Mathematics, University of Kentucky Press. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., and Krysl, P. (1996). Meshless Methods: An Overview and Recent Developments, Computer Methods in Applied Mechanics and Engineering 139, 3-47. Berenger, J. P. (1994). A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114, 185-200. Biro, O., and Richter, K. R. (1991). CAD in Electromagnetism, In Advances in Electronics and Electron Physics (P. Hawkes, Ed.), 82, 1-96. Birss, R. R. (1980). Multivector Analysis I-II, Physics Letters, 78A, 223-230. Bishop, R. L., and Goldberg, S. I. (1980). Tensor Analysis on Manifolds, New York: Dover. Bodenmann, S. (1995). Summation by Parts Formula for Noncentered Finite Differences, Research Report 95-07, Seminar ftir Angewandte Mathematik, ETH Ziirich. Boothby, W. M. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry, second edition. San Diego: Academic Press. Bossavit, A. (1998a). Computational Electromagnetism, San Diego: Academic Press. Bossavit, A. (1998b). How Weak Is the "Weak Solution" in Finite Element Methods?, IEEE Trans. Magn. 34, 2429-2432. Bowden, K. (1990). On General Physical Systems Theories, Int. J. General Systems 18, 61-79. Branin, F. H., Jr. (1966). The Algebraic-Topological Basis for Network Analogies and the Vector Calculus. Presented at the Symposium on Generalized Networks. New York: Polytechnic Institute of Brooklyn. Burke, W.L. (1985). Applied Differential Geometry. Cambridge: Cambridge University Press. Burnett, D. S. (1987). Finite Element Analysis, Reading, MA: Addison-Wesley. Cartan, E. (1922). Lefons sur les invariants intbgraux. Paris: Hermann. Cendes, Z. J. (1991). Vector Finite Elements for Electromagnetic Field Computation, IEEE Trans. Magn. 27, 3958-3966. Choquet-Bruhat, Y., and DeWitt-Morette, C. (1977). Analysis, Manifolds and Physics. Amsterdam: North-Holland. de Rham, G. (1931). Sur l'Analysis situs des vari6t6s /t n dimensions, Journ. de Math. 10, 115-199. de Rham, G. (1960). Varibtbs Diffdrentiables. Paris: Hermann. Deschamps, G. A. (1981). Electromagnetics and Differential Forms, Proc. IEEE 69, no. 6, 676-696. Dezin, A. A. (1995). Multidimensional Analysis and Discrete Models. Boca Raton, FL: CRC Press. Dolcher, M. (1978). Algebra Lineare, Bologna: Zanichelli. Eilenberg, S., and Steenrod, N. (1952). Foundations of Algebraic Topology. Princeton: Princeton University Press. Ferziger, J. H., and Peri6, M. (1996). Computational Methods for Fluid Dynamics. Berlin: Springer-Verlag. Flanders, H. (1989). Differential Forms with Applications to the Physical Sciences. New York: Dover. Fletcher, C. A. J. (1984). Computational Galerkin Methods, Berlin: Springer-Verlag.
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ADVANCES IN I M A G I N G AND ELECTRON PHYSICS VOL. 113
The Principles and Interpretation of Annular Dark-Field Z-Contrast Imaging P. D. NELLIST
1 AND
S. J. P E N N Y C O O K
2
1Nanoscale Physics Research Laboratory, School of Physics and Astronomy, The University of Birmingham, Birmingham, B15 2TT, UK. 20ak Ridge National Laboratory, Solid State Division, PO Box 2008, Oak Ridge, TN37831-6030, USA
I. I n t r o d u c t i o n . . . . . . . . . . . . . . . A. I n t r o d u c t i o n to A D F S T E M . . . . B. C o h e r e n t a n d i n c o h e r e n t I m a g i n g . . C. O u t l i n e . . . . . . . . . . . . . II. T r a n s v e r s e I n c o h e r e n c e . . . . . . . . A. S T E M I m a g e F o r m a t i o n . . . . . B. T h e C o n d i t i o n s for I n c o h e r e n t I m a ~,ing C. T h e R e s o l u t i o n L i m i t . . . . . . . . D. T h e T h i n S p e c i m e n O b j e c t F u n c t i o n E. D y n a m i c a l S c a t t e r i n g U s i n g Bloch W a v e s III. L o n g i t u d i n a l C o h e r e n c e . . . . . . . . . A. K i n e m a t i c a l S c a t t e r i n g . . . . . . . . B. D y n a m i c a l S c a t t e r i n g . . . . . . . . . C. H O L Z Effects . . . . . . . . . . . . D. T h e r m a l Diffuse S c a t t e r i n g . . . . . . . IV. T h e U l t i m a t e R e s o l u t i o n a n d the I n f o r m a t i o n L i m i t . . . A. U n d e r f o c u s e d M i c r o s c o p y . . . . . . . . B. C h r o m a t i c A b e r r a t i o n s . . . . . . . . . C. S o u r c e Size a n d the U l t i m a t e R e s o l u t i o n .... V. Q u a n t i t a t i v e I m a g e P r o c e s s i n g a n d A n a l y s i s A. T h e A b s e n c e of a P h a s e P r o b l e m . . B. P r o b e R e c o n s t r u c t i o n . . . . . C. D e c o n v o l u t i o n M e t h o d s . . . . VI. C o n c l u s i o n s . . . . . . . . . . . A. O v e r v i e w . . . . . . . . . . B. F u t u r e P r o s p e c t s . . . . . . . References . . . . . . . . . . . .
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147 Volume 113 ISBN 0-12-014755-6
ADVANCES IN I M A G I N G A N D E L E C T R O N PHYSICS Copyright 9 2000 by Academic Press All rights of reproduction in any form reserved. ISSN 1076-5670/00 $35.00
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A. Introduction to ADF S T E M The purpose of this paper is to describe how an annular dark-field (ADF) image is formed in a scanning transmission electron microscope (STEM), and to use that understanding to explain how the image data may be used to provide atomic-resolution information about the specimen. We start with a brief description of a STEM and the A D F detector. A STEM is in principle very similar to the more commonly known scanning electron microscope (SEM) in that electron optics are used before the specimen to focus an electron beam to form an illuminating spot, or probe (Figure 1), that is scanned over the specimen in a raster fashion (see Crewe [1980] for a description of the physics of STEM). Various signals produced by the scattering of the electrons can be detected and displayed as
FIGURE 1. Schematic of the scanning transmission electron microscope (STEM) showing the geometry of the annular dark-field (ADF) detector, and the bright-field (BF) detector for phase-contrast imaging.
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FIGURE 2. Simultaneously recorded (a) ADF and (b) BF images of G a A s ( l l 0 ) . The BF image shows lower resolution than the ADF image, and in this case the atoms are black contrast. The A D F intensity profile (c) shows the polarity of the lattice through the Z-contrast.
a function of the illuminating probe position. The major difference between SEM and STEM is that a thin, electron-transparent specimen is used in STEM, allowing transmitted electrons to be detected. Since there is very little scattering of the electrons in a thin sample, little beam spreading occurs and the spatial resolution is mainly controlled by the illuminating probe size. Typical electron-optical parameters for STEM are comparable to the conventional high-resolution transmission electron microscope, so that typical accelerating voltages are in the range 100-300 kV and the probe forming lens (known as the objective lens) has a coefficient of spherical aberration of the order of 1 mm. With such parameters, the illuminating probe can have a dimension similar to that of an atom (a few ~ngstroms) and atomic resolution imaging is possible. Annular dark-field imaging refers to the use of a particular detector geometry in STEM. A geometrically large annular detector is placed in the optical far field beyond the specimen (Figure 1). The total intensity detected over the whole detector is recorded and displayed as a function of the position of the illuminating probe. Since the detector only receives a signal when the specimen is present, the vacuum appears dark, hence the name, and the heavier the atom, the higher the intensity of the scattering, which leads to atomic number (Z) contrast in the image. The most important feature of ADF imaging is that it can be described as being incoherent, which has many advantages at atomic resolution. Figure 2 shows the incoherent
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and Z-contrast nature of an ADF image, and also how the phase contrast, bright field (BF) image can be recorded simultaneously for comparison. A major purpose of this paper will be to explain how the incoherence arises, why it is important, and how it may be used. The first step is to understand the difference between coherent and incoherent imaging.
B. Coherent and Incoherent Imag&g 1. High-Resolution TEM Hitherto, the majority of TEM imaging at atomic resolution has been performed using conventional high-resolution TEM (HRTEM) [Spence, 1988], and there is a huge range of applications (see Smith [1997] for a review). Conventional HRTEM is a coherent mode of imaging. The basic principle is that an electron transparent sample is illuminated by coherent plane-wave illumination. The exit surface wavefunction, ~(R), is then magnified by the objective lens, along with further lenses to provide additional magnification, to form a highly magnified image. Due to the inherent lens aberrations, especially spherical aberration [Scherzer, 1936], there is a blurring of the exit surface wavefunction as it propagates to the magnified image plane. This blurring can be written as a convolution with a point-spread function, P(R), and what is actually measured by a recording medium in the image plane is the intensity of this convolution, Icoh(R) = IP(R) |
~/(R)I 2
(1)
This equation is the mathematical definition of coherent imaging. The convolution of the exit-surface wavefunction with the point-spread function is in complex amplitude, which means that the scattering from spatially separated parts of the specimen can interfere in the blurring process. In practice, this coherent convolution means that the image intensity can fluctuate dramatically as the point-spread function is changed by changing the focus, for instance. Contrast reversals can occur depending on whether constructive or destructive interference is occurring, and there is uncertainty over whether atoms should appear as bright or dark contrast in the image [Spence, 1988], which can change depending on the focusing condition. In Figure 2, the atoms appear dark in the HRTEM image, but this is not always the case. The situation is further complicated by the existence of dynamical diffraction [Bethe, 1928], which can also have a strong effect on the coherent HRTEM image contrast, causing, for instance, contrast reversals as the thickness is changed (for example, Glaisher et al. [1989]). As the coherent
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electron wavefunction propagates through the crystal, the diffracted beams can be multiply scattered back onto one another leading to strong and complicated reinterference effects. In general, the exit-surface wavefunction cannot be directly interpreted in terms of the specimen structure, especially when defects such as grain boundaries are being studied [Bourret et al., 1988]. Generally, an approach is taken where images are simulated from trial structure models and are systematically compared with the experimental data [M6bus, 1996; M6bus and Dehm, 1996]. For this approach, the microscope imaging parameters need to be accurately known, and there is the ever-present danger that the correct structure will never be tried as a trial object or that an iterative process will find an incorrect local minimum. It is interesting to speculate as to why coherent imaging has hitherto dominated TEM. The reason may be partly historical and partly because of the instrumentation. Other than high resolution imaging, conventional TEM machines are also required to form diffraction patterns and diffraction contrast images. Both of these applications require coherent illumination to allow interference to occur to form the diffraction pattern. Before the advent of high-resolution imaging, these modes dominated the use of TEM. Additionally, HRTEM is a phase-contrast technique, which requires high coherence. As spatial coherence was lost due to the beam divergence increasing, users will have observed a loss in contrast, and the tendency would therefore have been to keep the spatial coherence as high as possible. The implementation of atomic resolution incoherent imaging has required the development of dedicated STEM machines capable of working at atomic resolution. Such machines have not been nearly as common as conventional TEMs, and thus most users have not had the chance to experience incoherent imaging.
2. Incoherent Imaging In his classic paper on the resolution limit of the microscope, Lord Rayleigh [1896] described the difference between coherent and incoherent imaging. He pointed out that if a transmission specimen was illuminated with light from a large source giving illumination over a wide range of angles, the specimen could be treated as being self-luminous, and that interference between the radiation emitted from spatially separated parts of the specimen could not interfere, and is thus incoherent. The image then becomes a convolution in intensity rather than in complex amplitude, I~n~oh(R) = IP(R)I2 | I~/(R)I2
(2)
which is the mathematical definition of incoherent imaging. Compare this with Equation (1). Lord Rayleigh also noted that the resolution limit of the
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microscope in the incoherent regime was twice that of a coherently illuminated microscope. The vast majority of images that we observe with our eyes are incoherent images, because most light sources are extended and incoherent. Lord Rayleigh pointed out that incoherent images do not show the sharp interference bands that characterize coherent images, and are therefore much simpler to interpret. It is because of their incoherent nature, that we are able to interpret the images that our eyes see. If we were surrounded by coherent illumination, the strong interference effects that we would observe would be very confusing. We might have to guess at what was near us, and perform an image simulation to compare with the image that our eyes observed! By remembering that the difficulties in conventional HRTEM, both the strong dependence on the imaging parameters and the problems of dynamical diffraction effects, are mostly born from the coherent nature of the illumination, it is clear that incoherent TEM imaging should hold major benefits. The first steps in this direction were taken in the development of annular dark-field (ADF) imaging in the STEM. An ADF detector had been used in the first STEM developed [Crewe et al., 1968a; Crewe et al., 1970] to collect a signal that was assumed to be largely elastically scattered. Because relatively high-angle scattering was collected by the ADF detector, the signal showed strong atomic number (Z) contrast, although the name Z-contrast was first applied to a signal derived from the ratio of the ADF to inelastic scattering. Later work showed that if any coherently scattered Bragg beams were incident on the ADF detector, the contrast could not purely be related to atomic number [Donald and Craven, 1979]. Various papers then suggested that increasing the angle of the ADF detector such that only incoherent thermal diffuse scattering, rather than coherent Bragg beams, reached the detector could ameliorate the coherent effects in the image [Treacy et al., 1978; Howie, 1979]. The steady improvement in STEM performance eventually resulted in the first ADF images being taken at atomic resolution [Pennycook and Boatner, 1988; Pennycook and Jesson, 1990; Shin et al., 1989]. These results, and Figure 2, illustrate how ADF STEM imaging gives a direct structure image with peaks at the atom sites, without the contrast reversals seen in coherent imaging. Since this initial work there has been a growing interest in the application and interpretation of ADF imaging, with numerous examples now in the literature. Inevitably, the physics behind ADF imaging is more complicated than simply asserting that incoherent imaging requires the collection of incoherent thermally scattered electrons; for example, the detector geometry itself can impose incoherence. The purpose of this article is to present the current understanding of the principles of ADF imaging, and the way the images may be interpreted.
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C. Outline
In this article, we have split the discussion of the destruction of coherence in the image into two components parallel and perpendicular to the beam direction, which we refer to as transverse and longitudinal coherence, respectively. In Section II we describe how it is the detector geometry that destroys the transverse coherence, even when purely coherent Bragg scattering is being collected by the ADF detector. This destruction of transverse coherence immediately allows direct interpretation of peaks in the image as being atoms or atomic columns in the specimen, which is extremely important for structure determination. In Section III it is shown that the detector geometry only weakly destroys coherence parallel to the beam, and that we must rely on phonon scattering to help remove longitudinal coherence effects. This is important in using the Z-contrast nature of the image for compositional determination, and we discuss the current state in this use of ADF images. Since ADF imaging is very different to conventional HRTEM, we need to examine the effects of other sources of incoherence, such as the source size and the chromatic aberrations. A treatment of such effects is given in Section IV, and we examine what might limit the ultimate ADF resolution. Finally, in Section V we describe how the ADF image might be used quantitatively to accurately determine atomic column positions using the incoherent nature of the image, before drawing some conclusions and putting forward some future prospects in Section VI. Throughout this article the Fourier transform of a real-space function, F(R), will often be written F(Q), where Q is a spatial frequency vector in reciprocal space. For consistency, the imaging parameters of a VG Microscopes HB603U STEM will be used in calculations. For this microscope, the accelerating voltage is 300 kV and the coefficient of spherical aberration, Cs, is 1 ram.
II. TRANSVERSE INCOHERENCE A. S T E M Imaye Formation
We start by developing a general formulation of imaging in a STEM that can be applied to imaging both elastic and inelastic scattering events. For elastic scattering, this formulation also holds for CTEM imaging through the principle of reciprocity [Cowley, 1969; Zeitler and Thomson, 1970], which allows the source and detector planes to be interchanged. The detector in the image plane of a C T E M is equivalent to the source plane in
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a STEM, and a certain detector geometry in a STEM corresponds to an illuminating source geometry in a CTEM. Thus a STEM can be thought of as a conventional T E M with the electrons propagating in the reverse direction. It is now clear that a small axial BF detector in a STEM (Figure 1) is equivalent to axial illumination in a TEM, and thus an H R T E M image will be formed with such a STEM detector. Unless explicitly stated, in this article we use the STEM point of view. A partial plane wave in the cone of coherent illumination is focused by the objective lens to form the illuminating probe (Figure 3). This partial plane wave can be labeled by the transverse component, K i, of its wavevector; by transverse we mean perpendicular to the optic axis of the microscope. Because of the objective lens aberrations, each partial plane wave will have experienced a phase shift, )~(Ki), relative to the K i = 0 wave, which in the case of spherical aberration
FIGURE 3. Contrast in a STEM image arises from interference between beams in the incident cone that are scattered into the same final wavevector and interfere. For Bragg beams, this interference can only occur in diffraction disc overlap regions.
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probe intensity profile
I t i -0.4
-
~.
~ , -0.3
~ t
~--0.2
-0.1
0
0.1
0.2
0.3
0.4
d i s t a n c e / nm FIGURE 4. The intensity profile of the illuminating probe at Scherzer defocus for the HB603U STEM.
and defocus is 1 3
z(Ki) -- ~z2zlK~l2 + ~2 CslKil
4
(3)
where 2 is the electron wavelength, z is the defocus, and C s is the coefficient of spherical aberration. Other terms, such as astigmatism and coma, could also be included in this phase function. An objective aperture, described by a circular top-hat support function H(Ki), is normally used to prevent strongly aberrated, high-angle beams contributing to the image forming process. The functions H(Ki) and z(Ki) can be combined as the magnitude and phase, respectively, of a complex aperture function, A(Ki). The illuminating STEM probe wavefunction, P(R) (Figure 4), is a sum over all the partial plane waves, P(R) = f A ( K i ) e x p [ - - i2rcKi 9R-] dK i
(4)
which we take as the definition of the inverse Fourier transform of A(Ki).
P. D. NELLIST AND S. J. PENNYCOOK
156
Scan coils are present in the STEM to move the probe across the specimen, and such a shift can be incorporated by multiplying A(Ki) by exp(i2rcKi-Ro), which gives P(R - Ro) when substituted in Equation (4). Consider now a scattering event that scatters from the initial partial plane wave, K i, into a final plane wave, with wavevector transverse component Kf, changing the magnitude and phase of the wave by a complex multiplier, q~(Kf, Ki). Assuming that there is no loss of coherence, we can write the intensity measured in the far field as I(Kf, Ro) =
If
A(K~)exp[-i2~zK i 9Ro]~(Kf, Ki) dK i
I
(5)
because Kf defines a position in the far field. Expanding the square gives a double integral I(Kf, Ro) =
' exp[i2rc(Ki-Ki)'Ro]~(Kf,Ki)tlfl*(Kf,K'i)dKidK'i f f A(K i)A *(Ki) (6)
However, this expression can be reduced back to a single integral by taking the Fourier transform of Equation (6) [Rodenburg and Bates, 1992] to give a function entirely in reciprocal space I(Kf, Q) =
f
A ( K i ) A * ( K i + Q)qJ(Kf, K~)qJ*(Kf, K i + Q)dK i
(7)
which is the Fourier transform of the image intensity that would be recorded for a point detector at position Kf in the far field. From Equation (7) it is clear that the contributions to an image spatial frequency, Q, come from pairs of incident partial plane-waves separated by the reciprocal-space vector Q (Figure 3). These two partial plane waves are scattered by the specimen into the same final wavevector, Kf, where they interfere. If the scattering is purely Bragg diffraction from a crystalline specimen, a series of Bragg discs will be seen in the form of a coherent convergent beam electron diffraction (CBED) pattern. In the disc overlap regions, two incident partial plane waves are being scattered into a single, final wavevector, and interference can occur. Since the probe position, R 0, is defined in reciprocal space by a linear phase variation over the incident wave, the phase difference between the two incident waves will be 2~zQ.R o. Scanning the probe, therefore, means that the interference between the two incident partial plane waves will cause the intensity to oscillate at a rate given by Q, leading to image contrast. Thus STEM lattice imaging depends on the detection of interference in the overlap regions of diffracted discs, as noted by Spence and Cowley [1978].
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B. The Conditions for Incoherent Imaging
In his discussion of incoherent imaging, Lord Rayleigh [1896] suggested that illuminating an object with incoherent illumination over a large range of angles rendered the object effectively self-luminous, and destroyed any interference between scattering from spatially separated parts of the object. Remembering that a large source in a conventional microscope is equivalent to a large detector in a STEM, we now need to include the effects of the detector. At this point it is easiest to consider a very thin specimen that can be treated as simply multiplying the illuminating wave by a complex function, O(R), that describes the magnitude and phase change of the transmitted wave. The Fourier transform of O(R) is written, T(Q). The Fourier transform of the image that would be recorded using an annular dark-field detector can now be formed simply by integrating Equation (7) over some detector function, D(K0, so that
I(Q) =
D(K0
A(Ki)A*(Ki+Q)~(Ke-Ki)~*(Ke-Ki-Q)dKi dKe
= f A(Ki- Q/2)A*(K~ + Q/2) x ~ D(K0~(K~- K~+ Q/2)~*(K ~- Ki- Q/Z) dKe dKi
(s)
J with the shift of K i by - Q / 2 allowed because the integral has an infinite limit. This is the expression derived by Loane et al. [1992]. The domain of integration over Ki is limited by A(K i -Q/2)A*(K i + Q/2) to be just the region of overlap between two objective apertures separated by Q. If D(K 0 has a geometry that is much larger than the objective aperture, the dependence of the Ke integral on K i becomes very small allowing the integrals to be separated, thus,
f
IADv(Q)---- A(K~--Q/2)A*(K~ + Q/2) dK~ • .f D(KOT(Ke- Ki + Q/2)'V*(Ke- K i - Q/2) dKe ,.,.,
- T(Q)O (Q)
(9)
where T(Q) is the transfer function for incoherent imaging. By analogy with light optics [Born and Wolf, 1980], this transfer function is referred to as the optical transfer function (OTF), and it contains information about the objective lens defocus, aberrations, and the aperture. Examples of OTFs for
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various degrees of spherical aberration have been calculated by Black and Linfoot [1957]. The function, O(Q) is the Fourier transform of the object function, containing information about the specimen scattering and the detector. We will examine its form for various specimen approximations later. Since Equation (9) is a product in reciprocal-space, taking its Fourier transform results in a convolution in real-space between the Fourier transforms of T(Q) and the object function spectrum, O(Q). The Fourier transform of T(Q), however, is simple to interpret. In Equation (9) it can be seen that T(Q) is the autocorrelation of A(Ki). The Fourier transform of the autocorrelation of a function is equal to the modulus squared of the Fourier transform of that function. Since the Fourier transform of A is, from Equation (4), the probe function, P(R), we can now write the ADF image intensity as
IADF(Ro) --IP(Ro)I 2 | O(Ro)
(10)
which is the incoherent imaging model as also given in Equation (2). So the image intensity can be straightforwardly interpreted as being the convolution between an object function and a positive-real point-spread function (PSF) that is simply the intensity of the illuminating probe. The convolution integral is, therefore, summing over the intensity of the probe, because interference effects between spatially separated parts of the probe are no longer observed, just as if the specimen were self-luminous. Hence the name incoherent imaging. The OTF, T(Q), is therefore simply the Fourier transform of the probe intensity function, ]P(R)]2. We have plotted the OTF at optimum defocus ( - 4 0 n m ) (Figure 5) compared to the conventional weak-phase object approximation (WPOA) contrast transfer function (CTF) for the same parameters. Using this thin specimen approximation, Equation (7) shows that the object function is O(Q) =
f
D(Kf)~(Kf-
K i @
Q / 2 ) ~ * ( K f - K i - - Q/2)dKf
(11)
where we require that the integral has no dependence o n K i t o allow the separation of the integrals in Equation (9). The domain of K i is a region defined by the overlap of two apertures separated by Q (Figure 3). If ~(K) is varying negligibly slowly over the scale of this domain, then the independence of Equation (11) on K~ will be fulfilled no matter what the geometry of the detector. A slowly varying ~(K) corresponds to a transmittance function, ~(R), that scatters from a highly localized source. Since the scattering is from a source much smaller than the probe geometry, interfer-
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ANNULAR DARK-FIELD Z-CONTRAST I M A G I N G
i 0.8 0.6
~F
0.4
0.2 0
........ p
-0.2
, 0.8
~
......
|
I
frequency ! (0.I nm) -i
-0.4 -0.6 -0,8
-! FIGURE 5. The optical transfer function (OTF) for incoherent imaging compared to the phase contrast transfer function (CTF) for the same defocus and spherical aberration. The radius of the objective aperture used for the OTF is marked.
ence between the scattering from spatially separated regions of the probe cannot occur and Equation (9) holds no matter what the detector geometry is. Treacy and Gibson [1995] have noted how a small source will give rise to an incoherent imaging model, but suggested that for more delocalized sources the model failed. However, for delocalized sources it is the detector geometry that can impose incoherence. If D(K0 allows the domain of integration of Kf to be much greater than the domain of Ki, then any variation of Equation (11) as a function of Ki will become negligibly small, vanishing completely ifD(K 0 is unity everywhere lade, 1977], which means that all transmitted electrons are detected. For a finite detector we are assuming that the detector is summing over complete overlap regions, an exact approximation in Figure 3, and that the contribution of any overlap region intersected by the edge of the detector is negligibly small compared to the rest of the detected signal. This, then, is the key to forming the incoherent model. For a given spatial frequency, Q, there are a range of pairs of incident partial plane waves that can contribute. For the incoherent model to apply, all of the available pairs must contribute to the image in a similar way so that a simple integral can be performed over these pairs as in Equation (9). To achieve this, a detector with a geometry much larger than a diffracted disc must be used. This
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criterion applies to both the detector and its inner hole. Hartel et al. [1996] have suggested that an inner radius at least three times the beam convergence angle should be used. Just as Lord Rayleigh proposed that a large source led to incoherent imaging in a conventional microscope, we have shown how a large detector leads to incoherent imaging in a STEM.
C. The Resolution L i m i t
One striking feature of Figure 5 is that the resolution limit for incoherent imaging appears to be twice that for the point-resolution in CTEM imaging. To understand the origin of this doubling of resolution, it is worth comparing incoherent imaging, with a large ADF detector, with conventional bright-field imaging that would be performed in HRTEM. Using plane-wave illumination in HRTEM is equivalent by reciprocity to only detecting the intensity at Kr = 0 in a STEM. Under these conditions, Equation (8) is no longer separable into two integrals, so the image intensity cannot be described by a convolution between an object function and a PSF as in the incoherent case. Substituting Kr = 0 into Equation (8) and inverse Fourier transforming gives the coherent imaging model in Equation (1). There are two important differences between Equations (1) and (10). First, Equation (1) shows that for coherent imaging the phase of the convolution between the probe function and the specimen transmittance, which are both complex quantities, is lost. If one wished to deconvolve the effect of the lens from the image, the phase problem would have to be solved first using holography [-Orchowski et al., 1995; Lichte, 1991] or focal-series reconstruction ECoene et al., 1992; Van Dyck and Coene, 1987]. Incoherent imaging does not suffer from a phase problem, and in principle the PSF can be deconvolved directly from the incoherent image described by Equation (10). Second, as seen in Figure 5, the resolution of the ADF is double that for the HRTEM image for the same objective lens and aperture. The origins of this effect can be seen by considering the coherent CBED pattern formed for two crystalline specimens with different lattice spacings (Figure 6). Interference can occur in the overlap regions between the discs, which depends on the phase difference between the diffracted beams, the lens aberrations, and the probe position [Nellist et al., 1995]. All STEM image contrast comes from such overlap regions. Conventional HRTEM imaging requires interference between the zero-order disc and two diffracted discs, and therefore the spacing in Figure 6(b) would not be resolved. However, an ADF detector will detect the interference regions for this crystal, and the spacing will be resolved in the ADF image. Here we
ANNULAR DARK-FIELD Z-CONTRAST IMAGING
FIGURE 6. For crystal (a) with a smaller detector senses
161
A schematic geometry of the STEM detectors relative to the diffracted discs. the BF detector senses interference between the 0, g, and - g discs. A crystal lattice spacing (b) gives discs more widely space, such that h > g. The BF no interference, whereas the A D F detector senses single-overlap interference.
have essentially restated, via reciprocity, the conclusion of Lord Rayleigh [1896] that the resolution is doubled for incoherent imaging. In real-space we can understand this effect in terms of the probe intensity profile (Figure 4) having approximately half the full width at half maximum of the probe amplitude profile because of the modulus squared being taken, thus improving the resolving power.
D. The Thin Specimen Object Function Using the thin-object approximation, we now consider the form of the object function, O(R), that applies for incoherent imaging. For illustration we first discuss an amplitude object, and then go on to the slightly more complicated case of a phase object.
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1. An Amplitude Object The theory we are developing is also highly applicable to energy-filtered imaging in a STEM, and to examine the effect of the collector aperture. For a general review of energy filtered imaging see Kohl and Rose [1985]. Let us consider a core excitation of a single atom. Our specimen function, ~p(R), will now have a magnitude that is localized in space. In principle we can collect all the electrons scattered by this excitation by using an infinite collector aperture. In this case D(K 0 is unity everywhere, and Equation (11) becomes the autocorrelation of ~(K), r
= f ~(Kf + Q/2)q'*(Kf - Q/2)dKf
(12)
which transformed into real-space gives an object function, O(R) = ]~(R)I 2
(13)
I(Ro) : IP(Ro)[ 2 @ I~P(Ro)l 2
(14)
such that
identical to Equation (2). We can see that a large collector aperture gives an image that approaches the perfect incoherent limit, as noted by Kohl and Rose [1985] and Browning and Pennycook [1995]. As the specimen function becomes more localized, as for higher energy excitations, we have seen that the detector geometry becomes irrelevant if the specimen function is much smaller than the probe dimensions. The effect of the collector aperture is not a simple convolution of the object function with the Fourier transform of the collector aperture. The transition from coherent to incoherent imaging as a function of collector angle, and the relative independence of localized excitations to the detector geometry, is all illustrated in Kohl and Rose [1985].
2. A Phase Object Let us now turn to the question of elastic scattering, which is the important process for Z-contrast imaging. Ignoring the effects of the electron wavefront propagation within the crystal, that is assuming a thin object, the specimen function can be approximated as a pure phase object (see, for example, Cowley [1992]), O(R) = exp[ - ia Vp(R)]
(15)
where Vp(R) is the projected potential integrated through the thickness of the crystal. The object function for perfect incoherence with an infinite
ANNULAR
DARK-FIELD
Z-CONTRAST
163
IMAGING
detector can be formed as it was for inelastic scattering by using Equation (13) which, since we have a pure phase specimen, gives unity everywhere and no contrast. This result is not surprising since no electrons are lost for a phase object, and therefore all the incident electrons will be detected. To form some contrast into the image, we need to introduce a hole in the detector, hence the use of an annulus in ADF imaging. The effects of the hole in the detector have been considered in the case of a weak-phase object [Jesson and Pennycook, 1993], but here we will generalize to a phase object which, while ignoring the effects of propagation, does include multiple scattering. Ignoring disc-overlap regions intersected by the inner radius of the detector, the object function can be written in reciprocal space, OaDv(Q) = f DADF(Kf)t[J(Kf + Q/2)t[t*(Kf - Q/2) dKf
(16)
Taking the Fourier transform of Equation (16) and a little algebra gives the real-space object function OADv(R) =
6(B) --
2-~Bi
0(R + B/2)0*(R - B/2) dB
(17)
where the term in parenthesis is the Fourier transform of an infinite detector minus a hole with radius, ui, J x is a Bessel function of the first kind, and B is a dummy real-space vector. Substituting the phase-object approximation, Equation (15), into Equation (17) and writing the integral over a half plane by symmetry gives OADF(R)
---
1 -- fh
Ja(2Tcu~[B[)cos{aVp(R + B/2) - alf plane
2~IBI
oVp(R
--
B/2)) dB
(18) Equation (18) shows that the hole in the detector has given rise to a coherence envelope given by the Airy function that is the Fourier transform of the hole. Within this envelope, the electron wavefunction scattered by spatially separated parts of the specimen can interfere, illustrated by the integral over the cosine function in Equation (18). If the projected potential, Vp(R), varies within the coherence envelope, then the cosine function will deviate from 1, and contrast will be seen. Although a finite-sized coherence envelope is required to form contrast, we are at liberty to make it as small as we desire, at the expense of reduced signal. For atomic resolution zone-axis imaging, we select an inner radius that results in a coherence envelope much narrower than the distance between neighboring atomic columns (Figure 7). Each column will then be acting as an independent scattering center, incoherent with its neighbors.
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P. D. NELLIST AND S. J. PENNYCOOK
j!(2mliB )
FIGURE 7. A schematic diagram showing the coherence envelope resulting from the hole in the detector and the projected potential. Ideally, the coherence envelope should be smaller than the interatomic distance, but to get contrast the potential must vary within the coherence envelope.
Equation (18) also shows that the object function is sensitive to the rate of change of the projected potential in the vicinity of R. The potential varies most quickly at the center of the atomic column giving rise to an object function peaked at the atomic column sites. The rate of change is strongly dependent on the nuclear charge, giving a sensitivity to atomic number, hence the name Z-contrast. If we assume that the specimen is thin and the phase shift small, then we can expand Equation (18) to second order in Vp(R),
OADF(R) = f Jl(2guilBI) 2rcIBI {r
+ B/2) - CVp(R - B/2)} 2 dB
(19)
cancelling the constants, which is the result derived by Jesson and Pennycook [1993]. An important difference between weak-phase contrast imaging in a CTEM and Z-contrast imaging of weak-phase specimens is that the lowest-order terms in Z-contrast imaging go as Vp squared, so it cannot be referred to as linear imaging. Linear imaging requires interference between scattered beams and the unscattered wave, whereas an ADF detector always detects interference between two scattered waves. Of course this effect does not hinder interpretability in any way, but can lead to effects such as an (002) spot present in the Fourier transforms of images of Si(110) [Hillyard and Silcox, 1995]. Although the projected potential of S i ( l l 0 ) does not contain an (002) Fourier component, the square of the projected potential will unless the potential is exactly a set of infinitely narrow 6-functions.
ANNULARDARK-FIELDZ-CONTRASTIMAGING
165
E. Dynamical Scattering Using Bloch Waves So far we have considered the destruction of transverse coherence for thin specimens, which do allow the inclusion of multiple scattering effects, but not propagation through the specimen. We must now consider whether we can use an incoherent imaging model in the presence of such propagation. To examine the effects of the detector alone, we will not yet include the effects of coherence loss due to inelastic scattering, for example, phonon scattering. We therefore need to use a theory of dynamical scattering with no absorption effects. We will use the approach of Nellist and Pennycook [1999], and assume a perfectly periodic specimen, so that we have strong Bragg diffracted beams incident on the detector. Ignoring the effects of higher-order Laue zone (HOLZ) scattering, which is found to have a negligible coherent contribution to the ADF detected intensity [Pennycook and Jesson, 1991; Amali and Rez, 1997], we can expand the solution to the full three-dimensional Schr6dinger equation, 4,(R,z), where z is the thickness, in terms of eigenfunctions of the twodimensional Hamiltonian (see, for example, Humphreys and Bithell [-19923 ~,(R, z) = Z ~0J)*~b(R) exp
J
-in
E(j)z~ -~o~j
(20)
where
hzV2
-- eV(R)
)
~b(J)(R) -- E(J)~b(J)(R)
(21)
and V(R) is the potential averaged along the beam direction. Since q~(R) must have the periodicity of the crystal lattice, it can be expressed in terms of Fourier components, ~J). By assuming plane-wave illumination, the boundary conditions give ~0j) as the coefficient of excitation of the jth Bloch wave in Equation (20). Using the Bloch wave formulation of dynamical scattering, we can write the ADF image intensity as I(Ro, z) = f DADF(Kf)
f A(Ki) exp[i2rcKi"Ro] ~ (I)(J)*(Ki)(I)~J)(Ki) Jg
I zE(J)(Ki)7 _]6(Kf--
x exp -ire
~
K i - g ) dKi dKf
(22)
where we connected Kf and N i via Bragg scattering through a reciprocal lattice vector g, the strength of which is given by the usual terms. Expanding the square in Equation (22) gives a double integral similar in form to Equation (6), but can be reduced to a single integral again by taking the
166
P. D. N E L L I S T
AND
S. J. P E N N Y C O O K
Fourier transform with respect to R 0 to give an entirely reciprocal space formulation, I-(Q, z) = ~ "og ADF f A(Ki)A(K i + Q ) ~ ,r,(J)*tK t ~ i/"~'g ~m(J)tv" ~,~(k)*,tC "x"0 ~, i]~ ( k Q )~l,n, l lX ij'a'g ~tx i) g
jk
x exp E- irc z(E(J)(Ki) - E(k)(Ki)) l dK
(23)
where we have now written the detector function in terms of the discrete Bragg beams that are detected, ignoring partially detected overlap regions. The advantage of using a completely reciprocal-space formulation is again clear. We can choose to evaluate the summation over the Bragg beams, g, first to examine the effect of the detector. The g summation acts only on a product pair of Bloch wave Fourier components resulting in
(24)
Cjk(Ki) : Z DADFO(gJ)(Ki)(~)gk)*(Ki) g so that Equation (23) can now be written I(Q, z) =
-
f
A(Ki)A(K i
+
Q)~ Cjk(Ki)~toJ)*(Ki)*~)(Ki) jk
x exp I_ irc z(EtJ)(K i)2E- ~Etk)(Ki)) 1 dK~
(25)
We can choose to evaluate Equation (24) using the real-space representations of the Bloch waves, and assuming the detector to be infinite except for a circular hole gives, Cjk(Ki)
= 6jk - f Jl(2~uiIBJ) 27~1BI f
r
C)q~tk)*(K i,
C
-~- B) dC dB
(26)
The multiplier containing the Bessel function is the Fourier transform of the hole in the detector, and is acting a kind of coherence envelope, controlling the degree to which different Bloch waves can interfere with each other. By introducing the detector function at this point we can determine which pairs of Bloch waves can interfere in the image forming process, simplifying any calculation by reducing the number of cross-terms that need to be considered rather than including them all and summing over the intensity incident on the detector, as has been used in multislice approaches [Kirkland et al., 1987; Hartel et al., 1996]. In examining the contributions of the various j, k terms to the imaging process, we will refer to the j = k terms as the "diagonal" terms, and the j 4: k terms as "off-diagonal." At zero thickness, when the exponential
ANNULAR DARK-FIELD Z-CONTRAST IMAGING
167
function in Equation (25) can be ignored, the orthogonality of the Bloch states means that the off-diagonal terms give an equal and opposite image to the diagonal terms, resulting from the fact that zero intensity will be scattered to the detector for a zero thickness specimen. As the thickness increases, the integration over Ki in Equation (25) starts to weaken the off-diagonal terms due to the exponential function and the dispersion of the eigenenergy of the states, and so an image starts to appear. At the limit of infinite thickness, any dispersion of the Bloch states will force the offdiagonal terms to zero, resulting in an image formed purely from the thickness independent diagonal terms, which we will refer to as the residual object function ROF. In this purely elastic and perfectly coherent description of scattering, the ROF is given by the sum of all Bloch waves weighted by their excitation coefficient and Cjj. In practice we arrange to use an inner radius such that the coherence envelope in Equation (26) is several times smaller than the smallest intercolumn distance that we wish to image. If the real-space form of the Bloch wave is slowly varying with respect to this coherence envelope, then the integral over C will approximate closely to being the inner product of ~b~J) with itself, which subtracted from the Kronecker 3-function will give a small value. Therefore, only Bloch waves that are peaked on a scale much smaller than the intercolumn distance will contribute significantly to the ADF images. A calculation of the imaging of I n A s ( l l 0 ) at 300kV using 311 Bloch waves (Figure 8) confirms this, showing that the R O F is dominated by the ls-type states, localized on the In and As columns, even though for the In columns the 2s state is the most excited. We have seen that a narrow coherence envelope exists controlling the Bloch wave interference, and this might suggest that an incoherent imaging model can apply here. The narrow coherence envelope ensures that only highly localized Bloch states can contribute to the ROF, and localized states are by their nature relatively nondispersive. Figure 9 shows how the ROF is largely independent of Ki, which again allows the integral over Ki in Equation (23) to be performed over just the aperture functions, as in Equation (9). Transforming back to real-space then gives the incoherent imaging model of a convolution between the probe intensity and the ROF. In addition, Figure 9 shows that the R O F displays strong chemical sensitivity, with the higher atomic-number column being the most intense. Both the insensitivity to K i and the chemical sensitivity can be explained physically by considering the high-angle limit of the inner radius. Nellist and Pennycook !-1999] have shown that in this limit the diagonal Cjj terms become proportional to the transverse kinetic energy of the state,
Cjj(Ki) oc <~bCJ)(Ki)[T]~bCJ)(Ki)>
(27)
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P. D. NELLIST AND S. J. PENNYCOOK
FIGURE 8. The residual object function (ROF) and its profile for lnAs at 300kV calculated using a 311 beam Bloch state calculation. The Bloch states summed over to form the ROF are shown, and illustrate how the ls states dominate the ROF. The ADF inner radius is 30 mrad.
T h e Cjj m u l t i p l i e r for the d i a g o n a l t e r m s is t h e r e f o r e j u s t the e x p e c t a t i o n v a l u e of t h e t r a n s v e r s e k i n e t i c e n e r g y for t h a t p a r t i c u l a r Bloch state. Since we are u s i n g a h i g h - a n g l e d e t e c t o r , we find t h a t o n l y B l o c h states with sufficient t r a n s v e r s e kinetic e n e r g y to s c a t t e r to the d e t e c t o r c o n t r i b u t e , so t h e d e t e c t o r is a c t i n g as a k i n e t i c e n e r g y filter with the l o c a l i z e d ls states
ANNULAR DARK-FIELD Z-CONTRAST IMAGING
169
FIGURE 9. The profiles over the dumbbell pairs of residual object function (ROF) for InAs at 300 kV as for Figure 8, but over a range of K i directions as marked. Note how the R O F hardly changes as a function of Ki.
dominating. So by forming an object function from the localized ls states weighted by their transverse kinetic energy, we obtain a map of atomic column positions weighted approximately by their atomic number, which we then image incoherently. An infinite detector with no hole would give perfect incoherence because the integral in Equation (26) would then reduce to 6jk, thus canceling out and destroying any interference between different Bloch waves. However, we mentioned earlier that at zero thickness the scattering by the off-diagonal terms was equal and opposite to the scattering by the diagonal terms, which implies that for an infinite detector, the R O F gives no scattering and, therefore, there is no image contrast. Just as for the phase object in Section II.D above, we need a hole in the detector to introduce some coherence to allow interference to create an observable from the wavefunction phase shifts that occur as the electron passes through the specimen. By increasing the inner radius of the detector we decrease the width of the coherence envelope, and therefore can approach the incoherent limit as closely as we like at the expense of reduced signal. To illustrate the effect of the hole in the detector, consider a situation where both the In and As columns are being illuminated by a large probe. Both the In ls state and the As ls state, denoted here by their subscripts 1 and 2, respectively, will be excited. The
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P. D. NELLIST AND S. J. P E N N Y C O O K
strength of the interference between the two atomic columns can be gauged from the C12 term, which we will evaluate here at K i = 0. Consider first a small axial detector that detects only one Bragg beam, such that only D o = 1. This is the situation encountered in conventional TEM imaging, since by reciprocity a small STEM detector is equivalent to a low-beam convergence in the TEM illumination. The coherence envelope, being the Fourier transform of the detector, is now relatively large and allows the columns to interfere. The ratio of the magnitudes of C I2(K i - 0 ) to C I~(K i = 0) is 1.25, and therefore the interference between the In and As columns is significant, varying as a function of thickness, and their contributions must be included in a coherent imaging model. If instead we use an annular dark-field detector with an inner radius of 30mrad (a typical experimental value), the ratio of the magnitudes of C I2(Ki = 0) to Cll(Ki = 0) falls to 0.02. Now the coherence envelope is much narrower than the intercolumn spacing, and the interference effects between the two columns is weak. Each column is being imaged as an independent scatterer, and an incoherent imaging model can be used. Thinking of the imaging process as being the detection of interference between overlapping discs in a coherent CBED pattern, intercolumn interference gives rise to features that are smaller than the geometry of the detector, and are therefore averaged out. Only features that are slowly varying over the detector, which come from the contributions from the individual columns, can contribute to the image-forming process. In this approach we have not considered the question of whether the probe channels down the atomic columns. Propagation of a focused probe through a crystal by a Bloch wave approach has been considered by Fertig and Rose [1981] and Pennycook and Jesson [1990], and more recently by a multislice method by Hillyard et al. [1993]. In the approach presented here, we have shown that it is the filtering effect of the detector that imposes incoherence. Only the ls states can contribute to the contrast, and these will only be excited when the probe is located over the corresponding atomic column. Beam spreading and the excitation of other states through the thickness is not important.
I I I . LONGITUDINAL COHERENCE
We have now discussed in some detail how interference between points that are separated in a direction perpendicular to the electron beam is not detected in A D F imaging, leading to an image that can be described by a convolution between the probe intensity and an object function. It has also
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been shown that the object function has a strong dependence on the atomic number, Z, of the species present in a atomic column. We have not yet, however, considered in detail the effects of interference between atoms within the same atomic column, referred to as intracolumn interference, which will affect the dependence of the intensity of the peaks in the object function on parameters such as the specimen thickness, or the position of a dopant atom within a column. For example, at the limit of high angle we might conclude that the Rutherford scattering from each atom is proportional to Z 2. If 10 atoms in a column are being imaged, is the A D F signal then 10 x Z 2 (which is the case for perfect incoherence along the column), or (10 x Z) 2 = 100 x Z 2 (the result if we assume that all the atoms interfere coherently and constructively)? In the latter case we would expect a quadratic dependence on thickness, whereas the former would give a linear dependence. In addition, conservation of energy gives a limit to the total intensity that can be detected, so the intensity cannot rise as a function of thickness forever. The phase object approximation of Equation (18) shows that at the limit of low thickness (that is low projected potential), the signal will rise quadratically as a function of thickness, as in Equation (19). It is clear that this has perfect longitudinal coherence. As the strength of the projected potential increases, the cosine function gives rise to an oscillatory dependence, which should be observable as thickness fringes. Such fringes are not observed in A D F imaging, so the longitudinal coherence is being broken to some extent. In this section we will first consider the role of the Fresnel propagation within the specimen, using the kinematical approximation. This approach is then extended to include the effect of phonons, and methods of including phonons in image calculations are listed. Finally, we discuss the effects of discontinuities in an atomic column, such as a single dopant atom or a surface adatom.
A. Kinematical Scattering
In Section II we saw how it was an integration in intensity over the transverse component of the scattering vector that destroyed the transverse coherence. Similarly, to destroy coherent interference within a column, known as intracolumn interference effects, we need an integration in reciprocal space in a direction parallel to the atomic column. For purely coherent scattering, this integration can only arise from an integration over the longitudinal component of the scattering vector, which comes from the curvature of the Ewald sphere. At high electron energies, however, the Ewald sphere is relatively flat and so the longitudinal components of the
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FIGURE 10. A schematic diagram showing the Ewald spheres for two partial plane waves in the incident cone, and the associated scattering amplitude for reciprocal lattice vectors with various excitation errors. The ADF detector sums over the Bragg beams from many such reciprocal lattice vectors.
scattering vector are small (Figure 10), so the detector geometry is much less efficient at breaking the longitudinal coherence than it is at breaking the transverse coherence. From Figure 10 it can be seen that the A D F detector will sum in intensity over different parts of the shape transform for different Bragg reflections. Kinematical theory (for details see Gevers [1970]) predicts that the amplitude of a reflection follows a sinusoidal dependence on specimen thickness, z, ~g oc Vgsin(rCSgZ)
(28)
gSg
and, therefore, the signal detected by the A D F detector as a function of thickness will consist of a sum over many of these sinusoidal oscillations, IADF(Z) OC~ D~IV.I z g
sinZ(rcSgZ)
(29)
7~ Sg
The thickness dependence will therefore depend on the structure and scattering factors of the specimen, and on the detector geometry. In general, the thickness dependence will start by showing an oscillatory behavior that progresses to a constant saturation value as the oscillations in the shape transforms of the reflections are averaged over by the summation over the detector [Treacy and Gibson, 1993]. The intensity oscillations can be
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interpreted as being the interference effects between atoms in the same column as the column increases in length, with the decay of these oscillations towards a mean value being the loss of coherence over a longer length column due to the detector geometry. Jesson and Pennycook [1993] point out that the first minimum of such oscillations will be close to the first minimum of the shape transform at a scattering angle given by the inner radius of the detector, which from Equation (29) is given by z - 22/02, assuming that the signal is dominated by scattering to this inner periphery of the detector. Their calculations for 100 kV ADF imaging of S i ( l l 0 ) with an inner radius of 50 mrad show the oscillations decaying over approximately 10 nm. From Figure 10, if Ki and Ki + Q are diametrically opposite in the cone of incident illumination, then the excitation errors for the interfering scattered Bragg beams, g and g + Q, will be equal. As K i varies over the incident cone, however, the excitation errors will become different for the two scattered beams, and the intensities of the scattering will vary as a result. Strictly, the integrals in Equation (8) can no longer be separated, and the image intensity can no longer be written as the convolution between a probe intensity and an object function. The explanation for this is that the Fresnel propagation of the probe through the specimen thickness changes the probe profile for scattering from different heights in the specimen, and there is not one single profile that can be applied to all the scattering [Treacy and Gibson, 1995]. We might expect this effect to become more pronounced as the thickness increases, but in that case a full dynamical scattering approach should also be used.
B. Dynamical Scattering The kinematical approach to the effect of thickness on ADF images has suggested that treating the image intensity as a convolution between the probe intensity and an object function should become less applicable as the thickness increases. Conversely in Section II.E it was observed that at the high-thickness limit, the object function was dominated by the bound ls states on the columns, and that the low dispersion of these states allowed the integral separation in Equation (25) giving rise to an incoherent imaging model. The multiple scattering gives rise to probe channeling [Fertig and Rose, 1981], which lessens the probe broadening effects. Having observed that the ls states dominate the scattering to the ADF detector, it is now instructive to examine the dependence of the dynamical object function given in Section II.E on thickness. Strong oscillations as a
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function of thickness are observed [Nellist, to be published], which appear to decay only very slowly and persist over many tens of nanometers. The frequency of these oscillations also depends on the atomic species present, and is controlled mainly by the eigenenergy of the ls state for that column. The effect is as though standing waves are formed in each column individually, with no crosstalk between the columns. Even though crosstalk may be occurring between adjacent columns, the destruction of intercolumn coherence by the detector prevents there being any effects from crosstalk in the ADF object function. The approach taken by Van Dyck and Op de Beeck [1996] is highly applicable here because of the ls state domination. The Bloch waves are written as a sum of the l s states on columns, n, bound to the columns and all the other states are collected as a background, ~(R, z) = ~ 4~,~S(R)exp n
- i~zzEXS1 2E ~ _] + ~
I
~b?)(R)
(30)
j:/: ls
When the phase of the ls state in a given column is close to the other less-bound states, the wavefunction is close to being a plane wave, and there is little scattering to the ADF detector, whereas out of phase, it gives a strong peak in the wavefunction, and therefore in the object function. Whereas the approach of Van Dyck relies on the eigenenergies to separate the ls states, here it is the ADF detector that is selecting the ls states, and therefore this approach can be applied over a much greater range of thicknesses.
C. H O L Z Effects In the kinematical and dynamical analyses given above, the effect of scattering to higher-order Laue zones (HOLZ) has been neglected. In practice, H O L Z scattering does occur at the high scattering angles detected by an ADF detector. Spence et al. [1989] pointed out that HOLZ scattering could destroy the incoherent imaging model by the introduction of phase contrast. The reason is that HOLZ scattering gives rise to very fine lines in the CBED pattern rather than discs, and thus only occurs for specific values of Ki. The effect is similar to using a small detector, and the integrals in Equation (8) are most definitely not separable. Measurements by Pennycook and Jesson [-1991] and calculations by Amali and Rez [1997] suggest that the contribution to the ADF signal by the HOLZ reflections is small compared to the thermal diffuse scattering (TDS) at those angles. Since TDS is a largely intrinsically incoherent scattering process, it is important to consider how it can affect the coherence in ADF imaging.
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D. Thermal Diffuse Scattering
It was suggested by Treacy et al. [1978] and Howie [1979] that in order to avoid coherent effects in an ADF image, the inner radius should be taken to an angle where thermal lattice vibrations have attenuated the strength of the coherent scattered Bragg beams, and the scattering is predominantly thermal diffuse scattering (TDS). In practice the situation is more complicated than this. In Section II it was shown that transverse incoherence occurs because of the detector geometry, even when it is detecting Bragg beams, but we might expect TDS to play a role in destroying intracolumn interference. It must be remembered, however, that lattice vibrations can be correlated between nearby atomic sites because the phonon spectrum has larger numbers of long-wavelength phonons, so we cannot assume that simply by detecting TDS, that all coherent effects will be destroyed. The situation is most easily visualized using the kinematical approximation. 1. TDS Usin 9 the Kinematical Approximation
The thermal vibrations of the lattice can be described in terms of the normal modes of vibration of the lattice. A quantum of excitation of the lattice is a phonon, so we may have many phonons in each mode. In addition to elastic scattering of the fast electron into the Bragg beams, there may well also be further scattering by a phonon. The phonon scattering can change both the momentum of the electron and its energy (see for further details Cowley [1975]. The latter effect renders the scattered electron incoherent with respect to the elastically scattered electrons. Thus the scattering to a final wavevector, Kf, can come from a region in reciprocal space given by the distribution of momenta of the scattering phonons (Figure 11). Over this region of scattering vectors, an integration in intensity occurs. Although this makes little difference to the transverse coherence because the detector is much larger than the range of phonon momenta (of the order of one reciprocal lattice), we now have an integration along the beam direction, destroying longitudinal coherence. Jesson and Pennycook [1995] have analyzed in more detail the extent to which the coherence is destroyed, but in general the loss of coherence is not as complete as in the transverse direction. 2. TDS with Multislice Calculations
It is now clear that to fully account for the effects of the partial longitudinal coherence, phonon scattering must be included in the dynamical scattering model. Calculations have been performed using both the multislice method and we will also discuss how a Bloch wave approach may be calculated using matrix diagonalization.
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FIGURE 11. The grey arrows depict scattering by phonons in addition to elastic scattering into a final wavevector, Kf. The phonons provide further integration in intensity in both the transverse and longitudinal directions.
The concept of the multislice calculation, proposed initially by Cowley and Moodie [1957] is straightforward. The crystal is divided into a number of thin slices. The projected potential within each finite slice is approximated to lie in an infinitesimal slice, thus acting as a thin-phase object, and the electron wavefunction is propagated between each slice as though it were in vacuum. For STEM, the incident wavefunction at the crystal is the probe's complex amplitude, so the calculation is more complicated than for plane wave illumination because a range of K i vectors are incident at the specimen. The first use of a multislice calculation for calculating STEM images did not include phonon scattering [Kirkland et al., 1987] and one application was the study of the visibility in ADF of single Au atoms on the surface of thin Si crystals [-Loane et al., 1988]. These early simulations suggested that the position of an adatom through the thickness of a crystal could change its resultant effect on the image contrast, complicating the interpretation of intensity. The multislice approach was then extended to include phonon scattering using the "frozen phonon" approach [Loane et al., 1991]. Since an electron accelerated through 100 k V has a velocity of about half that of light, the time taken for it to pass through a typical TEM sample is of the order of 10-16 s, about three orders of magnitude shorter than a typical phonon period of oscillation. The fast electron, therefore, can be regarded as interacting with a frozen "snapshot" of the crystal lattice. The dwell time of the probe at each probe position can be regarded as being long enough that many electrons pass through the sample, each one seeing a lattice with
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thermal atomic displacements uncorrelated with the displacements seen by other electrons. The recorded signal is the intensity of the electron scattering summed over many atomic displacement configurations. The accuracy of this approach was first checked by simulating convergent-beam electron diffraction (CBED) patterns that would be observed in the STEM detector plane [Loane et al., 1991], though the patterns simulated did not have overlapping discs so the beam convergence used would not produce a sufficiently small probe for lattice imaging in any STEM imaging modes. Having shown that the frozen phonon approach does reproduce many of the features observed in the TDS background, ADF STEM images can be simulated by calculating the coherent CBED pattern for each probe position, and summing in intensity over the region detected. Such calculations for atomic-resolution ADF imaging did indeed confirm that the image had transverse incoherence [Loane et al., 1992], and the behavior of the images as a function of thickness could also be studied [Hillyard et al., 1993; Hillyard and Silcox, 1993]. The calculations suggested that the intensity of an atomic column as a function of thickness did not show contrast reversals, such as those observed for coherent TEM imaging, but is not a linear relationship. The strong dependence on the atomic number, Z, was also shown so that heavier elements have more intense columns. An alternative approach to the inclusion of phonon scattering in a multislice calculation has been developed by Dinges et al. [1995]. Instead of displacing the atoms in the calculation, a calculation of the phonon scattering at each slice is made, and this TDS is propagated along with the rest of the electron scattering through the crystal to allow for multiple elastic scattering. However, the TDS is no longer coherent with the elastic scattering because of the inelastic nature of phonon scattering, so no interference effects will occur between the TDS and the elastic scattering. To incorporate this into the calculation, the phonon scattering at each slice is multiplied by a random "statistical phase" shift. The full multislice calculation is then repeated many times with different statistical phases so that any interference effects included by the calculation are eventually averaged over. This method has been applied in a practical way for the atomic resolution analysis of A1 concentration in A1GaAs quantum wells [Anderson et al., 1997], where 55 probe positions within a single unit cell were computed for various A1 concentrations, and compared with the experiment after substantial noise reduction procedures had been performed. It is worth emphasizing the point made by Hillyard et al. [1993] that a resolution limited lattice image consists of just a few complex Fourier components, and so can be completely characterized by just a few numbers. A simulation of a lattice image, therefore, only requires that number of probe positions in judiciously chosen positions to allow the entire image to be simulated.
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An approach using simple absorption potentials has been adopted by Nakamura et al. [-1997] for simulating single Au atoms substituted into a Si lattice. Further elastic scattering of the TDS giving rise to features such as Kikuchi lines is not included in this method, though since this only redistributes the TDS in scattering angle, and a large detector covering many scattering angles is used, it is interesting to consider how important this is. Since most of the absorption is TDS of which much will reach the ADF detector, it is important that this intensity is reintroduced to the final intensity measured in a way consistent with phonon scattering, which would not be peaked at zero scattering angle. Nevertheless, the calculations do suggest that "top-bottom" effects, with the intensity dependent on the Au depth in the crystal, will be observed. 3. Bloch Wave Calculations with TDS
Using the Bloch wave stationary solutions of the Schr6dinger equation to compute the electron wavefunction within a crystal has already been discussed and applied in Section II.E, though without any phonon scattering being included. It is possible to include absorption in such a calculation, and to also include an operator for phonon scattering, which operates on the electron wavefunction at all depths in the crystal to compute the TDS [Amali and Rez, 1997]. Such an approach does not include the effects of multiple phonon scattering, but the phonon scattering operator does allow for the creation (or destruction) of multiple phonons in one scattering event, so called "multiphonon" processes. The argument is made that subsequent elastic scattering need not be included if all the Bloch states are included. This argument only holds if all the phonon scattering is detected, which will only be true for a total detector. Physically this is neglecting any redistribution of the TDS from the detector into the hole, which will probably be a small effect for a large detector. Such calculations have been performed for perfect crystals [Amali and Rez, 1997] and stacking faults in crystals [Amali et al., 1997], showing that defects rather than composition changes can also lead to contrast in ADF images. Pennycook and Jesson [1990] have suggested that if we can assume that the ADF signal has been incoherently generated by phonon scattering at the atom sites, then a model can be used where Bloch waves are used to calculate the electron density at the atom sites, and the intensity at the atom site summed over thickness. It was shown that, for a probe located symmetrically at an atomic column, the l s states are predominantly excited, which is the origin of channeling. Neglecting the other Bloch states gives rise to a particularly simple thickness dependence. In Section II.E we saw that the ADF detector acts as a kind of Bloch state
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filter by selecting high transverse momentum, spatially localized states, which are the ls type states for zone axis imaging. Absorption and TDS by phonons also occurs over a highly localized region in the atomic column, and thus is strongest for electron wavefunctions with high transverse momentum, which is the origin of anomalous absorption. Redistribution of intensity by phonons from the Bragg beams to the TDS occurs for high-angle beams, and so selects the ls states in a similar way to the ADF detector. Note that it is not the phonon momenta that give rise to the high angles of TDS, since phonon momenta are typically of the order of a Brillouin zone, rather that the phonons preferentially scatter high transverse momenta electrons in the crystal. If we make the approximation that all the scattering incident on the ADF detector arises from ls states, either by coherent scattering or phonon scattering, then we can construct a simple prediction of the thickness behavior of ADF images. We now make an approximation, used by Van Dyck and Op de Beeck [1996] in the interpretation of conventional H R T E M images, that the eigenenergies of the non-ls states are negligibly small compared to those of the ls states. The wavefunction in the crystal is approximated as being the ls states, varying rapidly in phase through the thickness of the crystal, plus the non-ls states all propagating slowly, as described by Equation (30). When the ls and non-is states are in phase, as they are at the entrance surface, the wavefunction in the crystal will just be that of the incident wavefunction, which is the STEM probe and does not contain large enough momenta to reach the detector, so no intensity will be recorded. When the ls state is antiphase to the rest, the wavefunction will be the incident probe minus twice the ls state, so the electron density will show a large peak at the ls state and a large ADF signal will be recorded. This oscillatory behavior of the ls state can be seen in Bloch wave calculations [Fertig and Rose, 1981-1, and here we give its thickness frequency the symbol, ~. We now introduce the absorption from the ls state, with coefficient a, such that the coherent signal received by the ADF detector varies as /con(t) ~ e - ~ t [ 1
-
cos(r
(31)
where t is the specimen thickness. The absorption from the ls state may be integrated over thickness, if we assume it is perfectly incoherent between atoms, to give the total TDS O"
/,~s(t) oc (1 - e -~) -
1+
I
-~ e - ~ t
0 -2
sin(~t) + -~5 (1 -
e - o-t
cos(~t)) 1
(32)
The component of the wavefunction in the ls state can thus couple to
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FIGURE 12. A plot of the coherent, TDS and total intensities from Equation (33) using the parameters: a = 0.12 n m - l, ~ = 0.48 n m - l, and 0r = 0.17.
high-angle plane-wave states in the vacuum beyond the crystal either at the exit surface, where it emits as coherent Bragg scattering, or through phonon scattering. Let us assume that most of the TDS is detected, as it is biased to high angles, but much of the coherent scattering passes through the hole; the exact fraction depending on the size of the hole. We can define the fraction of coherent to incoherent scatter as c~ and write the total ADF intensity as
IADV(t ) = ITDS(t) + 0dcoh(t)
(33)
Equation (33) is therefore a three-parameter model with an analytical expression for the dependence of ADF intensity on thickness. The thickness dependence using Equation (33) with empirical values (Figure 12) compares very closely with the frozen phonon calculation of Hillyard et al. [1993] (see Figure 8 of that reference) for a column of In atoms. Of course, a full simulation is required initially to find the parameters and their dependence on column composition and the experimental conditions. Having parameterized ADF imaging in this way, however, may allow much
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faster inversion of intensity data to determine thickness and composition. Probably the most immediate benefit of this type of analysis is to confirm the dominating ls model, and to show that the image formation mechanism for ADF imaging is becoming understood, and is simple compared to conventional HRTEM imaging. Simplifying the image formation process opens opportunities for quantitative image analysis. Defects can cause further complications by causing interband transitions within the Bloch states and repopulating ls states that have been exhausted by absorption. An example of this effect is bright contrast arising from strain around B dopant atoms in Si [Perovic et al., 1993]. It is thus clear that interpreting ADF intensities beyond a qualitative approach still holds many challenges, but nevertheless, the strong Z contrast nature of the imaging is extremely useful.
IV. THE ULTIMATE RESOLUTION AND THE INFORMATION LIMIT
A. Underfocused Microscopy In Section II.C we have already discussed the resolution limit in terms of the intensity distribution of the illuminating probe, and in reciprocal space the transfer function decreases to zero at a spatial frequency given by twice the objective aperture radius. In conventional HRTEM, however, the form of the contrast transfer function can be modified by changing the degree of defocus, and pass bands can be pushed out in reciprocal space allowing higher spatial frequencies to contribute to the image (see for details Spence [1988]. The ultimate resolution limit of information that can be transferred by the microscope, or information limit, is then not defined by the spherical aberration of the objective lens, rather by other sources of incoherence, such as illuminating beam divergence, chromatic aberrations and the overall stability of the microscope. It is now interesting to consider how much of the principles of HRTEM imaging can be applied to ADF imaging. We start by showing that underfocusing the objective lens can counter the effects of spherical aberration leading to a resolution improvement, which then creates the question of what then defines the information limit in ADF STEM. Clearly we do not need to consider beam divergence since we have shown by reciprocity that ADF imaging is equivalent to using an extremely high beam divergence to destroy coherence. We must, therefore, go on to look at the effects of chromatic aberrations and the STEM electron source size, which can also destroy coherence. Since we are already describing the imaging as being incoherent, will these further sources of incoherence make any difference?
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s o u r c e / ' ~ _ ~~,,._
=
underlocus FIGURE 13. Spherical aberration causes overfocusing of the higher angle beams so that they cross over before the gaussian image plane. By underfocusing the lens, the higher-angle beams cross over at the specimen. It is these high angle beams that can carry high-resolution information.
Scherzer [1949] originally showed how, in the presence of spherical aberration, the image resolution could be optimized by underfocusing the objective lens. Geometric ray optics provides a simple way to picture this process (Figure 13). Spherical aberration can be described as being an overfocusing of the beams converging at higher angles, which is countered by an overall underfocusing. In terms of the incoherent OTF (Equation 9) it can be seen that underfocus is used to minimize the variation of the phase of A(Ki), which is Z, over the integral over Ki for all values of Q. Increasing the amount of underfocus will, in general, increase the phase variation and decrease the transfer. Increasing the underfocus, however, has pushed the turning point of Z out to higher values of Ki, which may be allowed to contribute if the radius of the objective aperture is increased. At certain values of Q that are beyond the usual resolution limit, the phase variation of A over the disc overlap region in (Equation 9) will become small (Figure 14) and the OTF will show a peak (Figure 15). Although the OTF does not now have the simple form of the Scherzer OTF, it does now extend to much higher spatial frequencies. The corresponding probe intensity distribution does show a smaller central maximum allowing higher resolution information to be passed, but at the expense of lower intensity and long, oscillatory tails (Figure 16). The lack of a phase problem in ADF imaging means that the probe can in principle be readily deconvolved, unlike the case in HRTEM where the phase problem must first be solved using, for instance, holography [Orchowski et al., 1995; Lichte, 1991]. The enhancement in resolution described also can be pictured by geometric ray optics (Figure 13): When the objective lens is highly underfocused, only the high angle beams are crossing at the specimen. These high angle beams are thus
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FIGURE 14. For large defocus, the turning point in the quartic form of g means that there will be some large Q where the integral across the overlapping discs in Equation (9) gives a peak in the transfer function. The attenuation by chromatic defocus spread is shown by the grey gaussian surface.
FIGURE 15. The OTF for a large objective aperture (17 mrad) and highly underfocused ( - 1 3 0 n m defocus) compared with that for Scherzer conditions (9.4mrad; - 4 0 n m ) . The underfocused OTF shows good transfer at high frequencies. The transfer at the origin is proportional to the area of the objective aperture.
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FIGURE 16. Probe intensity profiles for a large objective aperture (17 mrad) and highly underfocused ( - 1 3 0 n m defocus) compared with that for Scherzer conditions (9.4mrad; - 4 0 n m ) . Note how the central maximum has become very much narrower and the first minimum is less than 0.1 nm from the origin.
responsible for the sharp central maximum of the probe, and the out-offocus lower angle beams give the broader probe tails. Demonstration of resolution improvement as described has been demonstrated [-Nellist and Pennycook, 1998a]. In the ~112) orientation Si has a projected structure of pairs of atomic columns, separated by only 0.078 nm, arranged in a rectangular lattice. An ADF image recorded at Scherzer conditions using the VG Microscopes HB603U STEM (300 kV, Cs = 1 mm) is unable to resolve the column pairs, but using an objective aperture with a radius of 17mrad (approximately twice Scherzer) and a defocus of - 1 3 0 n m (Scherzer defocus is - 4 4 nm) gives an image where the column pairs are just appearing to be resolved (Figure 17) and the Fourier transform of this image shows that spatial frequencies as far as the {444} plane spacing are transferred, which is indeed enough to resolve the column pairs. Despite the probe intensity profile being somewhat complicated, the central maximum of the probe is still the strongest, still allowing direct interpretation to an extent, as shown in the simulation in Figure 18.
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FIGURE 17. (a) An image recorded with an objective aperture of 17 mrad and at approximately -130 nm defocus. The Fourier transform (b) shows transfer to the (444) planes with a spacing of 0.078 nm, resolving the pairs of Si columns, which can just be seen in the image and in profile plot (c) summed perpendicular to the pairs over 150 pixels. The profile plot also shows evidence of the probe sidelobes.
B. Chromatic Aberrations It is r e m a r k a b l e t h a t r e s o l u t i o n s well b e l o w a n 5 n g s t r o m h a v e b e e n a c h i e v e d in A D F i m a g i n g . I n H R T E M , i n f o r m a t i o n limits b e l o w 0.1 n m h a v e b e e n difficult to achieve. T h e l i m i t i n g f a c t o r is u s u a l l y the c h r o m a t i c a b e r r a t i o n s of the lens. E l e c t r o n s of different e n e r g y h a v e different focal l e n g t h s in t h e
FIGURE 18. (a) The object function for Si(112) showing the atomic column positions. The closest spacing between the columns is 0.078 nm. (b) A simulated image by convolving with the probe intensity shown in the underfocused case in Figure 16. Note how the atomic columns are resolved.
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FIGURE 19. The O T F with a defocus spread of 30 nm compared with no defocus spread and the chromatic envelope for HRTEM imaging for the same defocus spread. Note how the effect of chromatic defocus spread is much less severe for incoherent imaging, and just involves being limited in the midrange frequencies by an upper limit proportional to 1/Q.
objective lens, and the energy spread of the beam leads to an incoherent spread in defocus, A, and a coherence envelope in reciprocal space of the form [Wade and Frank, 1977] gchr (Q) = exp[- 89
]
(34)
This envelope sharply truncates the transfer of information in HRTEM (Figure 19), and is usually the major factor in controlling the information limit. Using tilted illumination can alleviate the effect giving rise to an "achromatic circle" in the transfer function [Wade, 1976] allowing certain spatial frequencies to be transferred at almost full strength. In this approach, however, only certain spatial frequencies are achromatic, and most are strongly attenuated by the chromatic defocus spread. Remembering that, from reciprocity, ADF STEM imaging is equivalent to HRTEM imaging with a large incoherent source providing illumination over many tilt angles, we may start to suspect that ADF imaging is robust to chromatic defocus spread since all spatial frequencies will have achromatic contributions. Indeed, it has been observed in calculations by Shao and Crewe [1987] that the intensity of focused electron probes is relatively
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insensitive to chromatic defocus spread. A simple quantitative approach to the probe broadening due to chromatic aberrations would be to add the diffraction broadening and chromatic broadening effects in quadrature to calculate the overall probe width. Here we can use the transfer function formulation of Section II.B to use a wave-optical approach to calculating the ADF chromatic envelope. By considering the transfer function strength calculated in Equation (9) as being an integral over overlapping microdiffracted discs, we can use the approach of Nellist and Rodenburg [1994]. The transfer function must be integrated over a defocus spread that is assumed to be a gaussian distribution
f expF_-z':V2/X2-1f H(K
i)H(Ki -
Q)
x exp[iz(Ki, z + z') - iz(Ki - Q, z + z')] dK~ d:~
(35)
where Z has been written as a function of defocus, z, explicitly. By substituting Equation (3), and noticing that Z is linear in z, some rearrangement gives Tchr(Q) = I t
exp[--z'2/2A2]
exp[ircz'2([Ki[ 2 -
[Ki -
Q[2)]
JJ • H(Ki)H(K i -
Q) exp[-ix(K~, z) - iz(Ki -- Q, z)] d~ dK i (36)
The ~ integral can be performed first, and is actually the Fourier transform of exp[-zZ/2A2], which gives another gaussian, Tchr(Q) = f
exp[- 89
.Q - IQI2)23
x H(Ki)H(K i - Q) exp[iz(Ki, z) - iz(K~ - Q, z)] dKi
(37)
after expansion of the moduli-squared and ignoring the overall scaling, which is of no significance here. From Equation (37) it can be seen that for incoherent imaging, the chromatic defocus spread does not lead to a simple multiplicative coherence envelope scaling the transfer function, as it does for HRTEM. The effect of the chromatic spread depends on the influence of the first exponential function on the K i integral. This influence is shown schematically in Figure 14. Along the line given by K i ' Q = IQ]2/2 there is no attenuation. The line is the perpendicular bisector of Q, and arises from interference between partial plane waves in the convergent beam that have the same angle with respect to the optic axis, and are therefore achromatic with respect to each other. Parallel to Q, there is an attenuation in the integrand of Equation
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(37) following a gaussian form that has a width that is inversely proportional to the defocus spread, A, and to the spatial frequency, Q. If the region of disc overlap that is the domain of the integral is narrower than the chromatic gaussian attenuation, then there will be little effect. It is only when the region of overlap, and therefore the transfer function, is large that the chromatic effects will be observed. Thus the effect of chromatic defocus spread is broadly to provide an upper limit to the transfer function. Since the width of the gaussian attenuation is inversely proportional to the spatial frequency, Q, the overall effect of chromatic aberration on the transfer function for incoherent imaging is to provide an upper limit that varies as 1/IOl. The above analysis is illustrated by some calculations. The approximately linear form of the Scherzer OTF (Figure 19) for the VG Microscopes HB603U is limited by the chromatic attenuation for a defocus spread, A = 30rim. Because the transfer function upper limit imposed by the chromatic dcfocus spread is proportional to 1/IQI, only the midrange spatial frequencies are affected. Counterintuitively, the highest spatial frequencies are hardly affected by the defocus spread. The (0.136 rim) -~ spatial frequency (required to resolve the Si< 110> dumbbell pairs) is only reduced by a factor of 0.75 for A = 30nm, whereas the attenuation of HRTEM given by Equation (34) for the same defocus spread is 4 • 10-17, which in practice means that no signal would be observed. This can be understood by realizing that the highest spatial frequencies in ADF arise from partial plane waves that must be close to being the objective aperture diameter apart, and are therefore constrained to be almost achromatic with respect to each other. To achieve an attenuation of 0.75 at (0.136nm) -1 in coherent imaging requires A = 2.3 nm, which is much harder to achieve experimentally. For the underfocused case illustrated in Figure 15, it is found that a defocus spread of A = 10 nm still allows sub~.ngstrom information to be passed at a reasonable strength [Nellist and Pennycook, 1998a-I, the 1/[QI upper limit being a much more slowly decaying function of Q than the sharp truncation of Equation (34). The robustness of ADF imaging to chromatic defocus spread has been explained above in terms of interference between partial plane waves in the convergent beam that are largely achromatic with respect to each other. It is also possible to explain it qualitatively in terms of the illuminating probe, since conventional phase contrast HRTEM images can be formed in a STEM by using a small detector. Coherent phase contrast images depend strongly on the phase variation across the coherent illuminating probe to provide the phase contrast, whereas incoherent images, such as ADF images, are insensitive to the phase and just depend on the probe intensity. The phase of the probe is a very sensitive function of defocus, as evidenced
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by the rapid contrast reversals that can be observed as the focus is changed, therefore integrating over a defocus spread is able to destroy much of the contrast in a phase contrast image. The intensity distribution in a probe varies more slowly than the phase, and thus the incoherent A D F image is much more robust to the integral over the defocus spread. C. Source Size and the Ultimate Resolution
It is clear from Section B above that incoherent imaging using A D F STEM is remarkably robust to chromatic aberrations, so we must consider other limitations to the resolution. We can regard the electron optics in a STEM as basically being present to demagnify the electron source such that it is imaged onto the specimen at atomic dimensions, so this demagnification and the effective source size is also important. The electron source can be regarded as being an ensemble of incoherent emitters distributed in space. Each emission point, after demagnification by the STEM electron optics, will give rise to a diffraction limited illuminating probe. The total illuminating probe intensity, J, must therefore be calculated by integrating over this ensemble, after taking into account the demagnification, J(R) = IP(R)I 2 | S(R)
(38)
where S is the geometric image of the source after taking into account the demagnification. Taking the Fourier transform of Equation (38) gives the O T F with a finite source, T~o~(Q) = T(Q)S(Q)
(39)
where S is the Fourier transform of the geometric image of the source. Now we can see that the finite source gives rise to a multiplicative coherence envelope on the transfer function and limits it. To have strong transfer at the highest spatial frequencies allowed by the transfer function, a geometric image of the electron source that is significantly smaller than the diffraction limit is required. In addition to the objective lens, a S T E M instrument is also fitted with condenser lenses, which allow the demagnification of the source to be continuously varied, so should not we simply use the highest demagnification possible? The answer is that we will have no current in the probe if we use this approach. In an optical system, the quantity known as the brightness is conserved, independent of the demagnification [Born and Wolf, 1980]. The brightness of a source is defined as the total current emitted divided by the emission and divided by the solid angle that the radiation is emitted into. In geometric ray optics, this quantity is conserved
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and at the probe we can calculate the total current in the probe as follows: The solid angle in the cone of illumination is held constant because of the objective aperture, whose radius is essentially defined by the spherical aberration. Thus the total current is proportional to the area of the geometric image of the source. As we demagnify to reduce the source size, we lose current. This current is being lost because the condenser lenses are more highly excited leading to the beam being more dispersed in angle, and thus being lost by the fixed objective aperture. It is now clear that an extremely high brightness source is required. Indeed the development of the STEM [Crewe et al., 1968a] required the invention of the cold field-emission gun (FEG) [Crewe et al., 1968b] to provide the required brightness. Given an electron gun of certain brightness, the operator has then complete freedom to trade geometric source size against probe current to optimize their experiment and the spatial resolution they require. It can be seen in Section IV.B above, however, that a 300 kV FEG has sufficient brightness for subfingstrom imaging to be achieved. The final limits to resolution that are likely to be encountered are those arising from mechanical and electrical instabilities, which, as in any high-resolution microscope installation, should be minimized as much as possible. Finally, we turn to the prospects for resolution improvement using spherical aberration corrector systems that are currently being developed [Haider et al., 1998; Krivanek et al., 1997]. If such a corrector corrects spherical aberration but not chromatic aberration, then it is clear from the analysis presented in Section B that it will have greatest application to incoherent imaging. As the spherical aberration is corrected, and the diffraction broadening of the probe is reduced, it will be necessary to demagnify the electron source further. It is important to note, however, that total probe current will not be lost because the corrected spherical aberration will allow the objective aperture radius to be increased, thereby exactly compensating. The current challenge appears to be the electrical stability required.
g . QUANTITATIVE IMAGE PROCESSING AND ANALYSIS
A. The Absence o f a Phase Problem
The mathematical formulation of incoherent imaging as being the convolution between an object function and the illuminating probe intensity has an important consequence that the probe intensity may be directly deconvolved from an image without the prior requirement to solve the phase problem. The implications of this on the way the image intensities can be
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analyzed in order to quantitatively determine atomic positions is discussed in this section. First it is necessary to understand the implications of the phase problem for deconvolutions in coherent HRTEM. The intensity in an HRTEM image can be written in the form of Equation (1). Both P and ~ are complex quantities, and by taking the final modulus-squared the overall phase of the convolution is lost. This so-called phase problem is important in many areas of diffraction and imaging [Burge, 1976] since it prevents direct inversion to the specimen structure. The probe complex amplitude, P, is only dependent on the microscope parameters and is, therefore, in principle, known. If the phase problem could be solved in Equation (1), then P could be deconvolved leaving the specimen function. Deconvolving P is thereby mathematically compensating for the effects of spherical aberration, and solving the phase problem in order to achieve this was the motivation behind the invention of holography [Gabor, 1948]. More recently, techniques such as off-axis holography [Orchowski et al., 1995; Lichte, 1991] and focal-series reconstruction [Coene et al., 1992; Van Dyck and Coene, 1987] have been applied to this problem. Even having determined, ~, it is still necessary to carry out relatively long computer simulations of the dynamical electron diffraction within the specimen [M6bus, 1996; M6bus and Dehm, 1996]. In Equation (2) we can see that there is no phase problem. The probe intensity is real and positive, and can be directly deconvolved. Our aim is to result in the ADF object function, which should consist of sharp peaks at the atomic column positions, the positions of which could be directly measured to quantitatively determine the atomic positions. The first step to achieve this is to compute or determine the probe intensity, P(R).
B. Probe Reconstruction
The probe function is purely a function of the microscope, and therefore a completely known parameter. In practice the parameters on which the probe function depends, such as defocus and astigmatism, will vary from image to image as, for example, the height of the specimen within the objective lens changes. In many images, however, the feature of interest is a localized defect, such as a dislocation core, located within a large region of perfect crystal of known structure. We can therefore estimate the object function in that perfect crystalline region and use the image data to determine the probe. A more detailed description of such a scheme is given by Nellist and Pennycook [1998b], but the basic ideas are as follows: The approach is to first determine the OTF for that particular image. The Fourier transform of such an image can be taken and the intensities of the
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corresponding spots measured. Making the approximation that the object function consists of 6-function-like peaks at the atomic column positions weighted by the square of the atomic number (Z 2) allows the Fourier components of the object function, or the incoherent structure factors, to be determined. These would be the expected spots strengths for a hypothetical perfect microscope with no resolution limit. Dividing the measured spot strengths by these computed Fourier components allows the microscope transfer function at these spatial frequencies to be determined. James and Browning [1999] have performed this procedure for different focus settings, and find that the determined transfer function values are close to those expected for the estimated focus. The observation can now be made that an image of a crystalline material does not allow the OTF to be uniquely determined, because only certain spatial frequencies are present. In general, perfectly periodic images contain very little information and can be characterized with only a few parameters. We are therefore to impose some form of constraint to proceed further. We assume that the transfer function is a relatively slowly varying function in reciprocal space, which is equivalent to assuming that we have a welllocalized probe in real space. A linear interpolation is therefore taken between the derived transfer values, which has been shown to be a good approximation [Nellist and Pennycook, 1998b]. The estimated transfer function can now be transformed back to reciprocal space to calculate the estimated probe intensity distribution. In the above example of Nellist and Pennycook [-1998b] using GaAs(110), the final estimated probe was found to be marginally broader than the optimum diffraction limited probe, which is to be expected taking into account the residual instabilities of a microscope.
C. Deconvolution Methods 1. Multiplicative Deconvolution In reconstructing the probe above, what we have done is deconvolved the object function from the image resulting in the probe intensity distribution. By dividing the Fourier transform of the image by the Fourier transform of the object function, we have performed a multiplicative deconvolution. A variety of different deconvolution methods exists [Bates and McDonnell, 1986], and we will explore the application of some of them to ADF object function reconstruction here, following the approaches of Nellist and Pennycook [1998b]. Having used a multiplicative deconvolution to reconstruct the probe from an image region containing perfect crystal, perhaps now the obvious place
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FIGURE 20. The simulated image from Figure 18 with the probe intensity having been deconvolved using: (a) a Wiener filter; (b) the CLEAN algorithm. Both these object functions are completely consistent with the experimental data.
to start is to apply a multiplicative deconvolution to the image area of interest, a defect say. Dividing the Fourier transform of the image by the transfer function will result in the Fourier transform of the object function. The difficulty is that the transfer function goes to zero at the resolution limit, and dividing by close to zero transfer values will dramatically amplify noise. To protect against this, a Wiener filter may be used (for a description see Bates and McDonnell [1986], which in our notation has the form I(Q)T*(Q) Orec(Q) = IT(Q)I 2 + e
(40)
~
where I is the Fourier transform of the recorded image, T* is the complex conjugate of the transfer function, and Orec is the reconstructed object function. A simulated image of Si(112) taken at Scherzer defocus is shown in Figure 18. The e parameter prevents an attempt to reconstruct the object function where the transfer is very low to minimize the impact of the noise. The result of applying a Wiener filter with e set to 10-3 of the maximum transfer value is shown in Figure 20. It is initially somewhat surprising that deconvolving the probe this way has made little difference to the image, and has not resulted in a sharply peaked object function from which we can read off the atom positions. The explanation of why the Wiener filter has been ineffectual lies in the form of the transfer functions. The initial O T F is a smoothly decaying function that goes to zero at the resolution limit. The Wiener filter results in a new effective transfer function that has a value of 1 until spatial
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frequencies where the transfer is weak and e starts to dominate in Equation (40). At this point the effective transfer function is decayed to zero. The result is an effective transfer function that goes to zero much more sharply than the original OTF. Sharp features in a transfer function lead to a degree of delocalization in the real-space image, and should be avoided. An optimum transfer function is one that decays in a smooth way to zero at the resolution limit, much like the original OTF. It is now clear that it is hard to improve upon the transfer function imposed by the microscope in A D F STEM, which also explains why A D F STEM images are straightforward to process and are often analyzed intuitively from the raw data or after a simple low-pass noise filter. It now becomes apparent that the obstacle to reconstructing a sharply peaked object function is not the form of the transfer function, rather the fact that information has been lost in the image-forming process. The object function contains information throughout reciprocal space, but above the resolution limit of the transfer function it is all lost. Since this information is thrown away, there can be many object functions that, once convolved with the probe intensity, can give rise to an observed image intensity. Such object functions would differ from each other in the information that exists beyond the resolution limit. To reconstruct to a single object function we need to impose a constraint further to the data we have measured in the microscope, using our prior knowledge and experience.
2. The C L E A N Alyorithm At this point it is worth mentioning an alternative approach to deconvolution, known as subtractive deconvolution. In attempting to deconvolve oscillatory point-spread function from astronomical radio telescope images, H6gbom [1974] invented an algorithm given the acronym, CLEAN. It has subsequently been applied to Patterson maps calculated from H O L Z electron diffraction patterns [Sleight et al., 1998], and here we examine its applicability to A D F images. The C L E A N algorithm has the following form: 1. Locate the pixel in the raw data that has the maximum intensity. 2. Transfer a fraction, 7, known as the loop 9ain, of the maximum intensity to the CLEANed reconstructed object function at that pixel location. 3. Subtract a PSF from the raw data, centered at the peak pixel position and with a height of 7 times that intensity. 4. Test the raw data to see if the contrast has fallen below some previously specified criterion.
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The results of applying the CLEAN algorithm to the image shown in Figure 18 is shown in Figure 20. It is clear that a much sharper object function has been reconstructed than that found using the Wiener filter. A closer inspection of the algorithm, however, shows that there will always be a spatial spread of pixel values reconstructed in the object function, and so single peaks at the atom positions will not be achieved. Furthermore, the sharpening of the object function shows that a constraint has been applied, but it is not clear what the nature of this constraint is. Tan [1986] has analyzed the CLEAN algorithm to determine the constraint, and found the algorithm broadly tends to reconstruct a "sharp" object function, which, of course, is ideal for atomic-like object functions. The mathematical details of the constraint, however, are found to vary strongly as a function of the loop gain, 7, and the stopping parameter. Since these will be varied depending on the noise in the raw data, it cannot be said that the CLEAN algorithm will quantitatively give a consistent result. Having said that, the fast and simple implementation of the CLEAN algorithm, and its reconstruction of peak objects, makes it an attractive proposition, and it may well be worthy of further investigation as a tool for ADF imaging.
3. Bayesian Methods The attempt in Section 1 above to use a Wiener filter to deconvolve the probe from an image demonstrated how it was not possible to determine a unique object function from the image data alone. The Wiener filter resulted in a blurred object function, which was a consistent object function for the image data, but was not the original one used consisting of an array of delta function. Both object functions are consistent with the data. To select between these valid object functions we need to incorporate some kind of constraint, but how should this be done mathematically? An approach that has been applied to image processing is to assert that the best we can hope to quantify is a probability distribution of the object functions given the experimental data (for example Sivia [1993]), written p(objectldata). The object function would allow us to determine both the most probably object function given the experimental data, but also allow us to quantitatively determine our confidence in that deduction [Skilling, 1998]. Using Bayes theorem, it is possible to relate this distribution to other probability distributions, p(object ] data) =
p(data I object)p(object) p(data)
For a noisy image, the probability of getting the measured data set for a
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given object function, p(data [ object), can be calculated by convolving the object function with the point spread function and performing a chi-squared comparison [Gull, 1978]. The probability of the data, p(data), is a constant, because the measured data is a constant. The probability of the object function, p(object), is the probability of that object function existing, without any other constraints. It is within this probability distribution that we must include our constraints, or prior knowledge about the system we are studying. It is therefore referred to as the prior probability distribution. Bayes theorem is making clear to us that there is no such thing as an unambiguous experiment. To interpret any experimental data, whether from an electron microscope or not, always requires prior experiences and knowledge. For ADF imaging, therefore, we need to find mathematical forms in which to encode our prior knowledge about the specimen. A popular prior in many different fields requiring image processing, for example radio, astronomy [Gull, 1978], has been entropy (for a review of applications see Buck [1991]). The prior has the form, p(object function)~ exp(~S) where S = - Z filog fi i
is the entropy, f~ are the pixel values for the object function, ~ is a constant. The idea is that entropy will always favor a smooth object; the highest entropy object is one that is completely uniform. Any structure in the object will result in a lower prior probability and will be less favored. Use of this prior in a Bayesian reconstruction will result in an object being reconstructed with the minimum structure consistent with the experimental data. An algorithm designed to locate the object function with the highest p(object [data) has been implemented by Skilling and Bryan [1984] and successfully applied to ADF data (McGibbon et al., 1994; McGibbon et al., 1995]. For precise quantitative applications, however, it has become clear that entropy cannot be used [Skilling, 1998]. A simple explanation is that the result obtained will depend on the number of pixels in the image, that is, on the sampling of the data. Clearly the object function should just depend on the specimen, not on the way that the data has been collected. Quantitatively correct prior probability distributions should fall into a specific class of mathematical functions [-Sibisi and Skilling, 1997]. Such priors do exist, and an extremely promising one is known as massive inference [Skilling, 1998], since it emphasizes objects with a small number of highly localized sources, ideal for atomic images.
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Clearly the lack of a phase problem in ADF imaging, and freedom from complicating effects such as contrast reversals, gives many opportunities for image processing and analysis. The vast majority of images are stil! analyzed in a qualitative way, but there is great scope for the development of quantitative methods that will provide investigators quickly and reliably with quantitative measurements of atomic column positions.
VI. CONCLUSIONS
A. Overview
What we have tried to show in this paper is that annular dark-field imaging in a scanning transmission electron microscope is simply the implementation in TEM of incoherent imaging as discussed by Lord Rayleigh [1896]. In general, incoherent imaging is far more common than coherent imaging. The images formed by our eyes are generally incoherent, since we usually have large incoherent light sources, and most optical microscopy is also incoherent. The great advantage to us of using incoherent imaging is that most of the interference effects that can complicate an image are destroyed, leading to much more straightforward interpretation of images. We would struggle to interpret what our eyes were seeing if we used mostly coherent sources, such as lasers, resulting in large fields of interference fringes. It is somewhat surprising that incoherent imaging in TEM has been so long in arriving, and must have much to do with requiring the microscope to form diffraction patterns and diffraction contrast images, both of which require coherent illumination. Such constraints do not apply at high resolution, and we hope w e have demonstrated that incoherent imaging holds many advantages. It must be acknowledged that incoherent image formation has lost information relative to coherent image formation. The use of a large detector geometry has averaged over many of the interference effects observable in the STEM detector plane. The complicated interference fringes seen in coherent imaging do contain information, but it is not easily invertible to a specimen structure. Generally the only way to proceed is to simulate images from trial structures to match against the images, with the danger that the true structure may be missed. The removal of information in the incoherent image formation process simplifies the interpretation of the data, and enough information is retained to determine the object structure. This allows the opportunity to observe unexpected structures, which is of crucial importance since it would be somewhat arrogant to assume that we could predict all the atomic structures that we are ever likely to observe.
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The type of information that can be derived from an ADF image falls into two broad classes: the determination of the projected position of atoms and atomic columns, and the determination of the elemental composition of the specimen. The former of these applications relies mainly on the destruction of transverse coherence, which results in the ADF image intensity being a convolution between the probe intensity and an object function with sharp peaks at the atom positions. Here we have shown that transverse coherence is destroyed by the geometry of the detector. Even if we assume purely coherent scattering, a large detector in a STEM is equivalent by reciprocity to a large incoherent source in conventional TEM, resulting in incoherent imaging. The destruction of transverse coherence is almost complete, and opens the way to a whole variety of image processing and analysis methods. The two-dimensional transverse nature of the ADF detector, while being highly efficient at destroying transverse incoherence, is much less efficient at destroying coherence between atoms along the beam direction. Here we must rely much more on the fact that at high angles we collect a large proportion of electrons that have been scattered by both an elastic scattering event and a phonon. The phase randomization by the phonon scattering can help in destroying the coherence between the scattering from different atoms, but the coherence destruction process is much less complete in this longitudinal direction. This residual coherence can be seen in effects such as strain contrast in the images, and provides to a limit to the degree to which the column intensities in an image can be quantitatively interpreted in terms of composition. The high angle nature of ADF imaging, though, does mean that the scattered intensities are a much stronger function of atomic number than in HRTEM, which relies on much lower angle scattering. This Z-contrast nature of ADF STEM is extremely useful and has found many applications, albeit generally of a qualitative nature.
B. Future Prospects
Although, to date, the number of machines worldwide capable of performing ADF STEM imaging has been rather low, there are already many examples of its application to materials problems ranging from semiconductor interfaces [-McGibbon et al., 1995] to supported catalysts [-Nellist and Pennycook, 1996]. The main hindrance has been that dedicated STEM instruments have been required, which have been very much research machines requiring dedicated staff. Recently, field-emission gun TEMs (FEGTEMs) have been displaying impressive capabilities as STEM instruments [James and Browning, 1999], thus combining TEM and STEM capabilities within one machine. With the prolific growth in the number of
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FEGTEMs being acquired, there will be a major growth in the use of the STEM approach, particularly ADF imaging, especially as the benefits of incoherent imaging are observed first-hand by a growing bunch of users. Naturally the growth in the number of applications of ADF imaging will promote further developments in the STEM capabilities of machines. Improvements may include better electrical and mechanical stability, improved scanning capabilities, and higher detector efficiencies. Such improvements will be of benefit also in the use of STEM for high spatial resolution microanalysis. In terms of image interpretation, the future lies in rapid quantitative deductions about the specimen. There is no doubt that qualitative interpretation of the image data will be the first approach used, as it is with most image data, but we have shown here that ADF imaging provides excellent opportunities for quantitative interpretation. Determination of atomic positions from the image data has been achieved, being relatively straightforward information to retrieve, and should quickly become fast and routinely done. Use of the image intensities for quantitative compositional determination is more difficult because of residual coherence, especially in the presence of, for example, strain. To date, attempts to perform quantitative determination have involved long calculations. Such calculations generally retain all the coherent scattering information, which is finally lost at the end when the summation over the detector is done, and therefore there is a lot of wasted time in the calculation. What is required are approaches where the incoherence is built in so that only the relevant terms in the calculation are retained. Further theoretical work is required here. More than one image may also be required for more detailed interpretation. For example, a series of images of the same area could be recorded over a series of different inner radii. This data set should be able to distinguish between the effects of strain and atomic number contrast. Finally, ADF STEM has already established itself as an important technique for the atomic-resolution investigation of materials, and will act as a catalyst for a much greater interest in all the applications of STEM.
ACKNOWLEDGMENTS
This work was supported by The Royal Society (London) and by the US DOE under Contract No. DE-AC05-96OR22464 with Lockheed Martin Energy Research. Some of the work was performed at Oak Ridge National Laboratory through an appointment to the ORNL Postdoctoral Program administered by ORISE.
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M6bus, G. (1996). Ultramicroscopy 65, 205-216. "Retrieval of crystal defect structures from HREM images by simulated evolution. I. Basic technique." M6bus, G. and Dehm, G. (1996). Ultramicroscopy 65, 217-228. "Retrieval of crystal defect structures from HREM images by simulated evolution. II. Experimental image evaluation." Nakamura, K., Kakibagashi, H., Kanehori, K. and Tanaka, N. (1997). J. Electron Microsc. 46, 33-43. "Position dependence of the visibility of a single gold atom in silicon crystals in HAADF-STEM image simulation." Nellist, P. D. and Pennycook, S. J. (1996). Science 274, 413-415. "Direct imaging of the atomic configuration of ultradispersed catalysts." Nellist, P. D. and Pennycook, S. J. (1998a). Phys. Rev. Lett. 81, 4156-4159. "SubSngstrom resolution by underfocussed incoherent transmission electron microscopy." Nellist, P. D. and Pennycook, S. J. (1998b). J. Microsc. 190, 159-170. "Accurate structure determination from image reconstruction in ADF STEM." Nellist, P. D. and Pennycook, S. J. (1999). Ultramicroscopy 78, 111-124. "Incoherent imaging using dynamically scattered coherent electrons." Nellist P. D. and Rodenburg J. M. (1994). Ultramicroscopy 54, 61-74. "Beyond the conventional information limit: the relevant coherence function." Nellist, P. D., McCallum, B. C. and Rodenburg, J. M. (1995). Nature 374, 630-632. "Resolution beyond the 'information limit' in transmission electron microscopy." Orchowski, A., Rau, W. D. and Lichte, H. (1995). Phys. Rev. Lett. 74, 399-401. "Electron holography surmounts resolution limit of electron microscopy." Pennycook, S. J. and Boatner, L. A. (1988). Nature 336, 565-567. "Chemically sensitive structure-imaging with a scanning transmission electron microscope." Pennycook, S. J. and Jesson, D. E. (1990). Phys. Rev. Lett. 64, 938-941. "High-resolution incoherent imaging of crystals." Pennycook, S. J. and Jesson, D. E. (1991). Ultramicroscopy 37, 14-38. "High-resolution Z-contrast imaging of crystals." Perovic, D. D., Rossouw, C. J. and Howie, A. (1993). Ultramicroscopy 52, 353-359. "Imaging elastic strain in high-angle annular dark-field scanning transmission electron microscopy." Rodenburg, J. M. and Bates, R. H. T. (1992). Phil. Trans. R. Soc. (Lond.) A 339, 521-553. "The theory of super-resolution electron microscopy via Wigner-distribution deconvolution." Scherzer, O. (1936). Z. Phys. 101,593-603. "~lber einige Fehler von Elektronenlinsen." Scherzer, O. (1949). J. Appl. Phys. 20, 20-29. "The theoretical resolution limit of the electron microscope." Shao, Z. and Crewe, A. V. (1987). Ultramicroscopy 23, 169-174. "Chromatic aberration effects in small electron probes." Shin, D. H., Kirkland, E. J. and Silcox, J. (1989). Appl. Phys. Lett. 55, 2456-2458. "Annular dark field electron microscope images with better than 2A resolution at 100 k V." Sibisi, S. and Skilling, J. (1997). J. R. Statist. Soc. B 59, 217-235. "Prior distributions on measure space." Sivia, D. S., David, W. I. F., Knight, K. S. and Gull, S. F. (1993). Physica D 66, 234-242. "An introduction to Bayesian model selection." Skilling, J. and Bryan, R. K. (1984). Mon. Not. R. Astr. Soc. 511, 111-124. "Maximum entropy image reconstruction: general algorithm." Skilling, J. (1998). J. Microsc. 190, 28-36. "Probabilistic data analysis: An introductory guide." Sleight, M. E., Midgley, P. A. and Vincent, R. (1998). Proc. EUREM-11, Dublin, 1996 2, 488-489. "Image improvement and information retrieval using the CLEAN algorithm." Smith, D. J. (1997). Rep. Prog. Phys. 60, 1513-1580. "The realisation of atomic resolution with the electron microscope." Spence, J. C. H. (1988). Experimental Hi qh-Resolution Electron Microscopy. OUP, New York.
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Spence, J. C. H. and Cowley, J. M. (1978). Optik 50, 129-142. "Lattice imaging in STEM." Spence, J. C. H., Zuo, J. M. and Lynch, J. (1989). UItramicroscopy 3i, 233-240. "On the HOLZ contribution to STEM lattice images formed using high-angle dark-field detectors." Tan, S. M. (1986). Mon. Not. R. Astr. Soc. 220, 971-1001. "An analysis of the properties of CLEAN and smoothness stabilised C L E A N - - s o m e warnings." Treacy, M. M. J. and Gibson, J. M. (1993). Ultramicroscopy 52, 31-53. "Coherence and multiple-scattering in z-contrast images." Treacy, M. M. J. and Gibson, J. M. (1995). J. Microsc. 180, 2-11. "Atomic contrast transfer in annular dark-field images." Treacy, M. M. J., Howie, A. and Wilson, C. J. (1978). Phil. Mag. A 38, 569-585. "Z contrast imaging of platinum and palladium catalysts." Van Dyck, D. and Coene, W. (1987). Optik 77, 125-128. "A new proceedure for wave function restoration in high resolution electron microscopy." Van Dyck, D. and Op de Beeck, M. (1996) Ultramicroscopy 64, 99-107. "A simple intuitive theory for electron diffraction." Wade, R. H. (1976). Optik 45, 87-91. "Concerning tilted beam electron microscope transfer function." Wade, R. H. and Frank, J. (1977). Optik 49, 81-92. "Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread." Zeitler, E. and Thomson, M. G. R. (1970). Optik 31,258-280, 359-366. "Scanning transmission electron microscopy."
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ADVANCESIN IMAGING AND ELECTRONPHYSICS,VOL. 113
Measurement of Magnetic Fields and Domain Structures Using a Photoemission Electron Microscope S. A. N E P I J K O ,
1 N. N. SEDOV, 2 AND
G. S C H O N H E N S E
3
Xinstitute of Physics, National Academy of Sciences of Ukraine, Pr. Nauki 46, 03039 Kiev, C.I.S./Ukraine 2The Moscow Military Institute, Golovachev str., 109380 Moscow, C.I.S./Russia 3Institut fiir Physik, Johannes Gutenberg-Universitgit Mainz, Staudingerweg 7, 55099 Mainz, F.R.G.
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Imaging of Ferromagnetic Domain Boundaries in a PEEM in the Operation . . Mode Without Restriction of the Electron Beam . . . . . . . . . . . . . . A. Formulas for the Image Calculation . . . . . . . . . . . . . . . . . . B. Image Calculation for Domain Boundaries of Various Kinds . . . . . . . C. The Numerical Calculation of Electron Trajectories in Strong Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Experiments on Observation of Magnetic Fields in P E E M Without Restriction of the Electron Beams . . . . . . . . . . . . . . . . . . . III. Imaging of Ferromagnetic Domain Boundaries in P E E M in the Case of Restriction of the Electron Beam by a Contrast Aperture or Knife-Edge . . . . A. Principle of Imaging of Magnetic Fields in the Case of Beam Restriction . . B. Estimation of the Image Contrast Due to the Local Fields . . . . . . . . IV. Magnetic Domain Imaging in X-PEEM Using Magnetic X-ray Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Principle of Element-Selective Magnetic Imaging . . . . . . . . B. Micropatterned Structures and Domain Walls . . . . . . . . . . . C. Exchange-Coupled Systems and Probing Depth . . . . . . . . . . . V. Magnetic Domain Imaging in UV-PEEM Using a Kerr-Effect-Like Contrast A. Magnetooptical Kerr-Effect and Its Manifestation in Threshold PEEM . B. First Detection of the Kerr-Effect-Like Contrast in UV-PEEM . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205 206 207 207 210 217 220 222 224 225 228 228 233 236 239 239 241 242 246
I. INTRODUCTION
A photoemission
electron microscope
(PEEM)
can be used for observation
o f t h e m a g n e t i c fields a b o v e t h e s u r f a c e o f a f e r r o m a g n e t . I n a c t u a l p r a c t i c e the potential observation
Volume 113 ISBN 0-12-014755-6
of PEEM of images
for the study of ferromagnets [Spivak
et al., 1 9 5 7 ] . I t p r o v e d
is f a r b e y o n d
only
to be feasible to
205 ADVANCESIN IMAGINGAND ELECTRONPHYSICSCopyright 9 2000by AcademicPress All rights of reproduction in any form reserved. ISSN 1076-5670/00 $35.00
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S. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE
perform quantitative measurements of fields on the ferromagnetic surface using the theory of image contrast [Dyukov et al., 1991]. Besides, the modern research methods with the use of the circularly polarized radiation enable realization of new PEEM capabilities for investigation of the magnetic domain structure of ferromagnets. Digital recording of the images obtained in PEEM allows to greatly increase the sensitivity of the device to relatively weak magnetic fields. In the present paper theory and experiments on study of the domain structure of ferromagnets using various operation modes of a PEEM are described. The calculation of the image contrast depth of the domain boundaries of different types is given. Experiments on application of the circularly polarized radiation for investigation of the ferromagnetic domains are presented. Possibilities of the quantitative estimation of the magnetic stray fields on the surface of objects under study are discussed. The present work may be of interest to specialists in electron microscopy, physical electronics, surfrace science, magnetism, as well as to undergraduates and postgraduates of the physical as well as physical engineering faculties of higher schools. II. IMAGING OF FERROMAGNETIC DOMAIN BOUNDARIES IN A PEEM IN THE OPERATION MODE WITHOUT RESTRICTION OF THE ELECTRON BEAM
In the general case the magnetic fields on the object surface can be observed in a PEEM due to two different reasons. First, in the case when the local photoelectron emission signal depends on the magnetization of the object region in question. Then special techniques, for example, photoemission under the action of circularly polarized radiation, are required (see below). Second, in the case when there is a local deflection of the electron beams during the initial section of their path away from the object surface. In Sections II and III we will consider the latter case. Two regimes of study can be considered: 1. The electron beams are not restricted by any apertures or knife edges on their way from the object to the microscope screen. 2. The electron trajectories are restricted by insertion of a diaphragm edge or other element into the beam path. The formation of image contrast caused by the local magnetic fields arises because of different reasons for the two operation modes. This results in different types of image contrast of the local magnetic fields and in different sensitivity of PEEM to these fields. In Section II the operation mode without restriction of the electron beams is described, in Section III the case,
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES
207
when the electron beams are restricted, is given. Let us consider the first case in more detail. The possibility of mapping of the electrical and magnetic fields on the object surface is connected with deflection of electrons by these fields above the object. Objects involving both electric and magnetic fields on their surface were investigated in a number of papers [Dyukov et al., 1991; Recknagel, 1941; Sedov, 1970]. From this point of view it is of interest to study the possibility of observation of ferromagnetic domain boundaries by the use of PEEM. Calculation of the image contrast due to the stray fields above the domain boundaries is presented below. The PEEM capabilities for study of the domain boundaries are also estimated. Let us consider several types of stray fields, which can exist above the domain boundaries of different kinds. 1. The field above the domain boundary expands outside vertically upwards above the boundary center and spreads symmetrically in a fan-shaped form on both sides. 2. 180~ (Bloch type) between in-plane magnetized domains. The induction vector of the magnetic field changes its direction along a spiral within this boundary. The induction directs along the surface within the domains. Above the boundary there is a perpendicular field component, which is maximum above the center of the boundary and decreases at its ends. Above the boundary there is also a parallel field component changing its direction on different sides of the boundary. 3. 90~ (N6el type). In principle, the magnetic flux of domains on the boundary of this kind may be closed within the ferromagnetic substance and does not expand outside above the surface (flux closure structure). However, if the angle between the boundary and the direction of magnetic flux within the domains is not exactly equal to 45 ~ then a stray field arises above this boundary. This field has the components both perpendicular to the boundary and parallel to it. 4. 180~ between domains in which the magnetic induction has components perpendicular to the object surface. The boundaries of this kind are inherent in Co because of its high anisotropy. In the following we derive the magnetic contrast in the image at the PEEM screen for all these cases.
A. Formulas for the Image Calculation Let us assume that the perfectly homogenous and plane object surface observed in PEEM coincides with the (x, y) plane of the Cartesian system.
208
S. A. NEPIJKO, N. N. SEDOV AND G. S C H O N H E N S E
The z-axis coincides with the axis of symmetry of the immersion objective lens. Above the object surface there is the homogenous electric field accelerating the electrons. Its strength is given by the following equation E o = Vo/l
(1)
where Vo is the accelerating voltage of the extractor electrode, and l is the parameter of the immersion objective lens equal to the extractor-cathode distance. Let us consider that the direction of the boundary between domains coincides with the y-axis of the coordinate system. Suppose that the electrons escape from the object surface with zero initial velocity, which is the limit of exact threshold emission. Then they are accelerated by the strong electric field in the perpendicular direction z. The calculation of the electron trajectories is made by the method of successive approximations. The zeroth-order approximation is the movement of electrons only under action of the accelerating electric field. Deflection of electrons from their perpendicular motion occurs due to the following interactions. 1. The perpendicular component of the electron velocity Vz interacts with the parallel components of the induction of the magnetic stray field B x and By. Under the action of the Lorentz force the electrons are deflected in the directions y and x, respectively. It is the first-order effect that leads to a shift of the point of electron collision with the screen and thus causes the image contrast. 2. The arising parallel components of the electron velocity, in turn, start to interact with the perpendicular components of the magnetic field B z, which results in the emergence of an acceleration in the directions x and y, respectively. This is the second-order effect. There are effects of higher orders, but they will not be taken into consideration in this paper because they are infinitesimal. The velocity, which the electrons attain in the parallel direction at the height z, can be calculated from the following considerations. The perpendicular velocity at the height z under the action of the accelerating field E o is equal to vz = x//ZeEoz/m
(2)
where e and m are the charge and the mass of the electron, respectively. The Lorentz force due to the magnetic field component B y ( z ) results in an acceleration of the electron in the direction of the x-axis. This acceleration is equal to e 5{ - -- Vz(Z)B y(z ) m
(3)
M E A S U R E M E N T O F M A G N E T I C FIELDS AND D O M A I N STRUCTURES
209
The integral action of the Lorentz force within the section from zero to the height z gives the electron velocity in this direction:
Vx(X, z) = emf ~ By(z) dz
(4)
Dyukov et al. [1991] have shown that due to the lateral electron motion of this kind the electron shift S~ at the microscope screen will arise. It is given by
Sx(x ) =
eEm~ fl X(z) dz
(5)
Because of the nonuniformity of this shift, the current density redistribution arises at the screen, that is, the image contrast is formed. Since the magnification of the uniform accelerating field is equal to 1, Sx refers directly to the sample coordinates. In a PEEM instrument the screen is placed behind objective and projective lenses, so that the corresponding magnification factors enter. A new function of the current density distribution on the screen is
jo(X) j(x + Sx) = 1 + dSx/dx
(6)
where jo(x) is the initial (zero-order) current density distribution before its redistribution by the Lorentz force. Substitution of expressions (3) and (2) into the formula (5) gives
Sx(x) = ~/-~o2e f l x/~ By(x, y, z)dz
(7)
This formula expresses the effect of the first-order infinitesimal under the action of the magnetic fields above the object surface. However, this effect manifests itself not for all kinds of magnetic fields. For example, if this shift arises along the y-axis owing to interaction with the field component B x and if the domain boundary is uniform and directed along the y-axis, then the redistribution of the current density at the screen does not occur. The reason is the equality of the shift values for every coordinate y. In cases of this kind, the image arises owing to effects of higher orders of infinitesimal. The effect of the second-order infinitesimal occurs due to the interaction of the tangential electron velocity given by Equation (4) with the perpendicular component of the magnetic field above the object. Equations (3) and (4) may also be written for the acceleration and electron velocity along the y-axis if the magnetic field component along the x-axis is taken instead of the component along the y-axis. In this case, after substitution of the
210
S. A. NEPIJKO, N. N. SEDOV AND G. S C H O N H E N S E
expression for this new component of the Lorentz force into Equation (5), it takes the following form
Sx(x ) =
Eo
vy(z)Bz(z) dz
(8)
where the velocity component along the y-axis should be substituted in the integral in the following form
vy(z) = _ _e I z Bx(z ) dz m do
(9)
Thus, the effect of the second-order infinitesimal is expressed by the double integral
Sx(x) = - ~ o
Bx(~)Bz(z ) d~ dz
(10)
From this formula the image will be calculated in the case when the first-order effect vanishes.
B. Image Calculation for Domain Boundaries of Various Kinds Case 1. For the domain boundary with the stray field, which spreads in the fanshaped form in the space above the boundary itself, let us approximate the distribution function of the magnetic induction just above the surface by the following expression B o ax Box(X ) = x2 + a2
(11)
Here B o is the maximum field value, and the value of a characterizes the half-width of the field distribution on the object surface. The perpendicular component of this field can be derived from Equation (11) by the Hilbert transformation:
Boz(X) =
Bo a2 2 a2 x +
(12)
The distribution of the field components in the space above the object is found as the solution of the Dirichlet problem for semi-space. This solution takes the form
B~ Bx(x , z) = x2 + (a + z) 2
(13)
MEASUREMENT OF MAGNETIC FIELDS AND D O M A I N STRUCTURES
211
Boa(a + z) B~(x, z) = x2 + (a + z) 2
(14)
and
The first-order effect does not exist for such a field, therefore, the image will be calculated from Equation (10). On substitution of Equation (13) in Equation (9), one can obtain the following equation for the parallel velocity component vy(z) = _e Boa
( a+z a tan ~
m
- a tan
a)
(15)
x
On substitutions, the integral expression (8) is calculated numerically. Obtained values of the shift function and data on its numerical differentiation are substituted in Equation (6). Results of this calculation of the current density distribution on the P E E M screen are plotted in Figure 1. The electron current density at the screen as a function of the x coordinate in the direction perpendicular to the domain boundary is demonstrated in this figure. The value of a characterizes the boundary half-width. As is seen from the figure, the domain boundary of this type will be imaged a t the
J/Jo 1.5
1.0
0.5
,
-15
.
,
-10
-
.
.
.
.
,
.
-5
.
.
.
.
0
.
.
.
5
b..
10
x/a
FIGURE 1. Plot of the current density distribution on the P E E M screen calculated for the domain boundary of the first type. The characteristic length (a) denotes the half-width of the field distribution of the object surface.
212
S. A. NEPIJKO, N. N. SEDOV AND G. S C H O N H E N S E
microscope screen as a dark line. The contrast depth is strongly increased here for clearness. In fact the contrast is very weak for realistic magnitudes of the magnetic induction and of the half-width of its distribution. A quantitative estimation of the decrease of the current density in the central part of the boundary image comprises a value of the order of 10-5. This is a very low contrast. Nevertheless, the modern digital methods, recording to the brightness distribution function by means of a cooled charge-coupleddevice (CCD) camera, along with the statistical methods of data processing, permit picking out such a signal on the image. Case 2. The magnetic field distribution for the Bloch-type domain 180 ~ boundary of the second type with a screw-shaped turn of the magnetization vector within the boundary will be approximated by an equation similar to Equation (11), but now it is taken for the field component directed along the y-axis
Boax Bor(X ) = x2 + a2
(16)
For a field of this kind the first-order effect already manifests itself. On substitution of the spatial field distribution corresponding to Equation (16) in integral (7), it is calculated analytically. However, the resulting expression is rather complicated, and it is not given here. The plot of the current density distribution is constructed again from the calculated values of the shift S and of its derivative. It is interesting to note that the boundary image depends on the rotation sense of the magnetization within the Bloch wall. If the magnetic induction vector changes along the direction perpendicular to the wall, twisting as a left-handed screw, then the boundary is observed as a bright strip. Again, the absolute value of contrast is very small; it is of the same order of magnitude as in the first case. The line-scan plot of the calculated current density for this case is presented in Figure 2. If the turn of the induction vector is a right-handed screw type, the boundary is seen as a dark strip. The plot of the current density for this case is shown in Figure 3. Case 3. The magnetic flux of the N6el-like 90~ is closed within the substance, but a part of the flux emerges on the surface because of a deviation of the angles between the magnetization direction and the boundary direction. Then the magnetic field component perpendicular to the boundary gives only the first-order effect calculated already in case 1. However, the field component arising due to the field asymmetry and directed along the boundary manifests itself in the form of a stronger effect of the first-order. For this case, a turn of the bell-shaped magnetic induction
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES
213
~0 1.5
...............................................................................................i~0 ............................................................................................................
0.5
T
-15
T
, . .
-10
.
,
.
.
.
.
-5
0
.
.
5
.
.
10
x/a
FIGURE 2. Plot of the current density distribution on the P E E M screen for the domain boundary of the second type with rotation of the magnetization vector of the left-handed screw type.
J /0 1.5
1.0
0.5
-15
-10
-5
0
5
10
x/a
FIGURE 3. The case similar to the one shown in Figure 2, but the turn of the magnetization vector is of the right-handed screw type within the boundary.
214
S. A. NEPIJKO, N. N. SEDOV A N D G. S C H O N H E N S E
on the object surface is approximated by the following equation
Boa2(a 2 - x 2) Bor(X) =
(a 2 + x2)2
(17)
In this case integral (7) is also calculated analytically, but the obtained expression is again rather complicated.
S(x)=a2Bo 2~/~o [ ( 2 x21~
In a+x//2alxl+'xlx/x2_k_a 2
a
tan
x/ZalaXl)
a3/2 -]
+ x2+a2j
ixi -
(18) The plot of the corresponding current density distribution on the microscope screen is shown in Figure 4. In this case the boundary is imaged as an asymmetric double strip: a bright strip is accompanied by a dark one parallel to it. The depth of contrast for the fields of this kind may be much higher than in the preceding cases. It can reach 1%. Case 4. 180~ with emergence on the object surface not only of the magnetic flux existing within the boundary, but also of the main
~0
1.0
0.5
-3
-2
-1
0
1
2
x/a
FIGURE 4. Plot of the current density distribution at the PEEM screen for the domain boundary of the third type in the case of the asymmetrical N6el-type 9ff~-boundary. The maximum brightness on the screen is followed by a minimum. The depth of contrast is overstated for clearness.
M E A S U R E M E N T O F M A G N E T I C FIELDS AND D O M A I N STRUCTURES
215
magnetic flux of the domains themselves. It is inherent in materials with strong anisotropy, for example, for Co. The fields of domains forming a characteristic pattern emerge on the hexagonal surface. This pattern was observed not only by powder figures, but by PEEM as well [Spivak et al., 1957]. If the angle between the object surface and a prismatic crystal face is small, then domains in the form of parallel stripes (striped domains) are observed. Above the domains there are magnetic fields expanding outside. In these cases very high image contrast is observed owing to the great values of the magnetic stray fields and their wide spread. Let us calculate the image of this type in PEEM. The domain structure will be considered to have a periodic character with the period d along the x-axis. The periodic function of the magnetic induction distribution on the object surface may be expanded into a Fourier series. The influence of field harmonics decreases in proportion to the period of the harmonics as the height above the object increases. Therefore, at sufficiently great height the first harmonic, corresponding to the period d, will dominate. Let us calculate the image for it. The distribution of the parallel and perpendicular components of the magnetic field just above the object surface will be assumed to take the following form: 2nx Boz(X) = Bo sin - - ~
2nx Box(X ) = - B o c o s - - ~
(19)
The following functions of the spatial field distribution correspond to these equations
Bz(x'z)=B~ Bx(x, z) =
sin2nxd
B o exp
-
(20)
cos-j-
The image is calculated by Equations (8) and (9) for the second-order effect. On substitution of function (21) in Equation (9), one obtains
vy(x,z) = B~
exp ( - ~ f ) l
sin 2nX--d-
(22)
Then Equation (8) leads to the expression
BZod2e 4nx Sx(x) - 16n2Eom Sin--~ -
(23)
The period of this function is half of the initial one, which can be explained by the independence of the effect on the magnetic field direction.
216
S. A. N E P I J K O , N. N. SEDOV AND G. S C H O N H E N S E
J~0 3.0
2.0
t. . . . . . .
0
,.......
0.5
1.0
1.5
x/a
FIGURE 5. Plot of the current density distribution at the PEEM screen corresponding to the image of strong stray fields of domains for substances with strong anisotropy like in Co.
The fact that B o enters to the 2nd power testifies to the second-order magnitude of this effect. Under substitution of realistic parameters for this case one can obtain that the depth of contrast at the screen may be rather large, so that even caustics may be observed at the screen. The image calculated for a set of parameters is presented in Figure 5. Thus, this is the only case when the stray-field induced image of ferromagnetic domains at the microscope screen yields a very high contrast. However, it should be noted that in this case not the domain boundaries, but the stray fields of domains themselves are observed. Despite the high contrast it is complicated to interpret this type of image, which does not reflect unambiguously the domain boundaries. The question now arises whether it is possible to observe the domain boundaries themselves without strong distortion. To answer it, let us make an additional calculation. The derivative of the shift function (23), on which the depth of contrast depends, is
B~de Sx(x) = - - c o s - 4nEom
4nx d
(24)
Therefore, the condition of weak contrast when there is no strong distortion
M E A S U R E M E N T O F M A G N E T I C F I E L D S AND D O M A I N S T R U C T U R E S
217
of the image takes the form
B2de E~ >> 4rc------m
(25)
It means that for an experiment of this kind it is necessary to choose a sufficiently strong field accelerating the electrons.
C. The Numerical Calculation of Electron Trajectories in Strong Magnetic Fields When the condition of weak image contrast is not met, the first approximation used above does not give the exact picture in the image calculation, but only an approximate one. Then the numerical calculation of the electron trajectories in the complicated combination of the electrical and magnetic fields can be used. An advantage of this calculation is the possibility of predicting the image on the microscope screen when there are fields of any form and intensity. The drawback is the fact that this calculation is correct only for a particular type of field; the calculation should be made again under change of any parameter. Another drawback is the impossibility of the analytic solution of the inverse problem on the image contrast, that is, the calculation of the magnetic field from its image. In this case it is only possible to compare experimental and calculated pictures and to select parameters used in the calculation in such a way that both images are similar. Let us give another example of the numerical image calculation for the case of strong magnetic stray fields for objects of the Co type. The magnetic field above the object surface can be approximated by the equations
Bx(x, z)
Bo in (L + z) 2 -Jr- X 2 =-7(a+z) z +x 2
B=(x'z) 2B~ rt a tan (
-- a tan
(L+Z))x
(26) (27)
Here a is the half-width of the domain boundary, L is a parameter having dimensionality of length equal to the object thickness (for example, the thickness of the Co crystal), B o is the value of the perpendicular component of the magnetic induction originating from the domains themselves. The construction of the image of this field shows that it reflects the real field above the domain boundary of Co type objects rather well.
218
S. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE
z, [am
z, ILtm
20
20
0
0 0
100
200
y, gm
-2
-1
0
1
x, gm
FIGURE 6. Electron trajectories in the ZOY (a) and ZOX (b) planes starting from a region of strong tangential magnetic field near the domain boundary of a ferromagnet with strong magnetic anisotropy of the Co type. Parameters--see in text.
The accelerating electrical field of the microscope immersion objective E o acts on electrons leaving the object surface. The calculation of electron trajectories in this complicated field was made by the Runge-Kutta method with the adaptive choice of the step of integration. It is known that in a uniform electrical field and in a magnetic field normal to it the electrons move along trajectories in the form of a cycloid. In our case the magnetic field is not uniform, but if its induction is sufficiently large, the electron trajectories will be similar to a cycloid. They come back again to the object surface after each loop. The typical behavior for this case can be seen in Figure 6. The calculation was performed for the following parameters: B 0 =0.9T, a = 10-8 m, E o = 4.106 V/m, the initial electron velocity was taken equal to zero. The coordinate of the starting point of the electron is assumed at a distance of 1/~m from the center of the domain boundary. In this region the parallel component of the magnetic field is rather large, therefore, the electron trajectories in the (y, z) plane are close to the cycloid form. In the (x, z) plane electrons move almost periodically along the loop-shaped trajectories. Thus, electrons cannot leave the central boundary part at all, where there exists the strong tangential field. It means that the corresponding regions will look like dark spots or stripes at the microscope screen. The trajectory of an electron moving in the same field is shown in Figure 7, but here the point of emission is at a distance of 2.05/~m from the domain boundary center. In this place the parallel component of the magnetic field is less than in the previous case. Therefore, an electron, which performed a rather complicated motion in both planes, may now escape and move towards the anode. Here an intermediate case of the character of motion is realized. However, the escaping electron acquires a rather high tangential velocity; its value and direction significantly change at a slight change of the
219
MEASUREMENT OF MAGNETIC FIELDS AND D O M A I N STRUCTURES
z, ~tm
z, ~tm a
200
200
/
0 ~" 0
50
y, ~ m
-5
0
x, n m
FIGURE 7. Electron trajectories in the ZOY (a) and ZOX (b) planes starting from a region of a magnetic field of average value in the case when the emission point of the electron moves away from the domain boundary.
conditions of escape. Hence, electrons of this kind produce an additional background of scattering at the microscope screen; they do not take part in the formation of a well-defined image. Thus, electrons with trajectories of the types shown in Figures 6 and 7 promote formation of the domain boundary image as a dark stripe. At last, if an electron escapes still farther from the center of the domain boundary, its motion becomes already ordered. This case is shown in Figure 8. Here the initial coordinate of the emission point of the electron is 5/~m, consequently the magnetic field above the object is still weaker. If one traces the electron trajectory of this kind up to the microscope screen, then it will occur that the electron shifts in the lateral direction from the center of the
z, lam
z, ktm t/
/
40 20
/
40
/
.
b
/ J
20 ----__.___
0
. . . . . .
0
1 O0
200
0
y, lam
- 10
-5
0
x, n m
FIGURE 8. Electron trajectories in the ZOY (a) and ZOX (b) planes in the case when the emission point is far away from the domain boundary.
220
S. A. N E P I J K O , N. N. SEDOV A N D G. S C H O N H E N S E
j(x)/Jo ! 1.0 J
f f
f
0.8 0.6 -40
-20
0
20
40
x, btm
FIGURE 9. Plot of the current density distribution at the microscope screen when imaging a domain boundary of Co.
domain boundary image. Hence, the domain boundary of this kind will look like a dark stripe with a high image contrast. Let us consider now the case with the magnetic fields of other character. If the perpendicular component of the magnetic induction above the surface has a value, at which electrons from the domain boundary region do not return to the object, the picture is not so well defined. The boundary also looks like a dark strip, but the contrast decreases although it remains sufficiently deep. A case of this kind may occur if, for example, the macroscopic object surface is set at an angle to the hexagonal plane of the object crystal lattice. The numerical calculation of the current density distribution for such a case is given in Figure 9. Here the maximum value of the perpendicular field component in a region far from the boundary was taken equal to 0.3 T. The other parameters have the same magnitudes as in the previous example. It is seen from the figure that at the screen the boundary looks like a dark stripe with easily observable contrast. Its depth reaches about 15% in the center. At these conditions the boundary width being visible at the screen is about 40/tm. Electrons shifted from the boundary center are grouped together at a distance from the boundary, therefore, weakly pronounced maxima of brightness are observed there. D. Experiments on Observation of Magnetic Fields in P E E M Without Restriction of the Electron Beams As mentioned above, this method of observation of magnetic fields using P E E M was experimentally realized for the first time by Spivak et al. [1957].
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 221
FIGURE 10. Image of magnetic domains on the hexagonal face of a Co single crystal obtained in the PEEM without restriction of the electron beam.
The magnetic stray fields on a hexagonal face of a Co single crystal were investigated. Since Co has a strong magnetic anisotropy perpendicular to this plane, the strong magnetic fields of domains forming a specific pattern emerge on the hexagonal face (Figure 10). This image was obtained in a simple P E E M model with a magnetic lens. The object was illuminated by visible light, which could not cause direct photoelectron emission from Co. Therefore, in order to obtain the photoelectron emission, a thin layer of stibium-caesium (CsSb) photocathode with low work function was deposited on the object surface. The Co single crystal being studied was a cube with a 5-mm edge. Upon the image formation this cube was placed in an external magnetic field. It was generated by the microscope magnetic lens and amounted to 0.07 T. As was shown in Section II of this paper, dark regions on the image correspond to regions with the strongest magnetic stray field emerging on the object surface. It is interesting to note that when the external magnetic field reverses its direction, the image contrast changes its sign. The same surface region of the hexagonal face of a Co single crystal is shown in Figures 1 l(a) and (b) for comparison. These images were obtained in P E E M without restriction of the electron beam (Figure l l(a)) and by Bitter technique (Figure 11 (b)). The object imaging with Bitter-pattern is the classical method for the study of domain boundaries. The image presented in Figure l l(b) was obtained by deposition of a thin suspension of iron
222
S. A. NEPIJKO, N. N. SEDOV AND G. SCH()NHENSE
FIGURE 11. Domain images on the hexagonal face of a Co single crystal: (a) Image obtained in the PEEM without restriction of the electron beam; (b) Bitter-patterns obtained for the same object region.
oxide (Fe304) , prepared in a chemical way, on the object surface. In this case the object was kept in an external magnetic field of the same value and direction. The conformity of the patterns of the domain structure in both images is clearly seen. According to the above mentioned, it is difficult in this way to calculate the distribution of the magnetic induction of these strong fields above the object surface due to the complicated shape of the electron trajectories forming the PEEM image. We could have performed the described experiments without lowering the work function by adsorbates if the PEEM had been equipped by a CCD camera having an amplification factor as high as 10 6 and if, for example, a Dz-high-pressure tube with an excitation energy up to 6.3 eV had been applied for excitation.
I I I . IMAGING OF FERROMAGNETIC DOMAIN BOUNDARIES IN P E E M
IN THE
CASE OF RESTRICTION OF THE ELECTRON BEAM BY A CONTRAST APERTURE OR KNIFE-EDGE
The technique of imaging the magnetic stray fields emerging on the object surface using partial restriction of the electron beam enables the observation
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 223 E K
L ,-,~,,,:,:,,U,~::,:,:,,: ~
.
.
-..
.
I
.
"...........~.X.. :.'~.
'
b
'
-
'
'
7 Z
"::.:... : . . . ~
x I
_
__ f
~!
"
~
.....~:~..._ ~
" "'~'~. ."~'~w.
FIGURE 12. Principle of the image contrast formation of the magnetic fields in PEEM due to restriction of the electron beam. K--object surface, L--microscope cathode lens, C-diaphragm restricting the beams, E--microscope screen, a--electron beam leaving an object point without its deflection by local magnetic fields, b--midline ray of the same beam deflected by the magnetic field.
of much weaker fields. This is because the P E E M sensitivity to the field can be significantly enhanced. Such a technique is described by Dyukov et al. [1991], Sedov [1970], and Mundschau et al. [1996, 1998]. The principle of formation of this type of contrast is exhibited in Figure 12. The electron beams leaving the object surface are deflected by the magnetic field. In this case the magnitude of the beam deflection itself is not important, but the additional tangential electron velocity component, which is acquired due to the local magnetic field. The deflection of the beam center from the normal trajectory during its movement in an electron-optical system occurs just at the expense of this velocity. If on the electron-optical path part of the beam is obscured, then the intensity of the beam passing further depends on its deviation. Depending on the direction of the beam deflection, object regions with magnetic fields above them will look either darker or brighter than the background. This is similar to obtaining dark-field or bright-field images in light optics using the T6pler method (Schlieren method). As is seen from Figure 12, the optimum position of a diaphragm restricting the beam is the crossover plane C. Then the contrast caused by the magnetic fields will be the same all over the field of view, and sensitivity to the local fields on the object will be maximum. However, to do this, an aperture diaphragm or a plate, which can be shifted, should be placed in the
224
S. A. N E P I J K O , N. N. SEDOV AND G. S C H O N H E N S E
crossover plane of the emission microscope. Its shift is required in order to obtain the maximum image contrast that occurs under blanking of one-half of the undeflected beam. Such a microscope construction is described by Sedov et al. [1962].
A. Principle of Imaging of Magnetic Fields in the Case of Beam Restriction Let us consider the electron trajectories in PEEM in the case when there are local fields on the object surface (Figure 12). Electrons escaping the surface of object K are accelerated by the uniform electric field. The lens of the microscope immersion objective focuses an enlarged image of the object at screen E. In the microscope the projective lenses can also exist, which additionally magnifies the image. They are not shown here for the sake of simplicity. This is of no significance because they simply transfer the image to the screen with a selectable magnification, without changing the image itself. The trajectory of electron a escaping from the object normally to the surface plane is shown by a solid line. In reality a whole beam of electrons, exhibited in the figure, leaves the same point, and they have various directions of movement and a nonvanishing starting energy. Assume that in the crossover plane C (back focal plane of the objective) there is the aperture diaphragm with a hole. This diaphragm is shifted relative to the optical axis as shown in the figure. A knife-edge partially obscuring the electron beam can be placed instead of the diaphragm. Then a part of the electron beam with the midline of the trajectory a will be cut off, and the image brightness at the screen will be decreased. If the electrons in the beam are deflected by the action of the local fields above the object, their trajectories are shifted in one direction or another. The trajectory of the central electron of the deflected beam b is presented by a dashed line. The brightness of the corresponding region at the screen depends on the direction of deflection and can increase or decrease. If this deflection occurs due to the local magnetic field above the object surface, the brightness depends on the direction, intensity, and spread of this field. Let us suppose that this deflection in Figure 12 is directed upwards, then the corresponding region of the object surface will be brighter than adjacent regions (the dark-field image). If the beam is deflected in the opposite direction, this region will look darker than adjacent regions (the bright-field image). Thus the image of these fields can be seen at the screen. Let us calculate the depth of image contrast for such a method of observation of fields. The electron velocity acquired in the tangential direction by the local magnetic field is given by Equation (4). Since the local fields do not spread
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES
225
very highly, infinity can be substituted for the upper limit in this equation. If the value of By(x, y) is expressed as the solution of the Dirichlet problem by the magnitude of the magnetic fields on the object surface itself Boy(X, y), one can obtain the following equation (these calculations are given by Dyukov et al. [1991])
y) =
e f~oof~o B ~
+ ,)2
(28)
At the expense of this acquired velocity in the crossover plane C of the immersion objective, the electron gets a lateral shift
fVx
s x =x/2evo/m
(29)
where f is the focal length of the immersion objective (Figure 12). In this plane the electron beam has a current density distribution close to a Gaussian shape owing to the initial velocities of electrons escaping from the object:
Jc = Jco e x p ( - rZ/p 2)
(30)
where r is the distance from the axis, p is the effective radius of the distribution function:
p = Uw/VT/Vo
(31)
VT is a potential corresponding to the mean energy of escape of electrons from the object. This magnitude for photoelectrons can range from tenths of volts to several volts. The relative value of the electron deflection under the action of the local fields is of importance for the image contrast. This value can be obtained after combining Equations (28) and (31): ,,
Sx(x,y)
Sx
1 k / 2 e f~o~f ~_ B o r ( X - ~ , y - q ) d~drt = p = 2--~ mYT oo N / ~ 2 nt- t~ 2
(32)
B. Estimation of the Image Contrast Due to the Local Fields Equation (30) along with Equation (32) enable calculation of the image contrast formed due to the local magnetic fields and estimation of the maximum sensitivity of the PEEM to these fields. To do this, assume that in the crossover of the microscope immersion objective there is an aperture diaphragm of a diameter much less than the value of p. Then the value of
226
S. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE
FIGURE 13. Image of magnetic domains on the prismatic face of a Co single crystal obtained under restriction of the electron beam by the aperture diaphragm. The diaphragm is located in the crossover plane according to Figure 12.
the electron current passing to the microscope screen from each object region will be proportional to the value ofjc according to Equation (30). In this equation the sum of the initial shift of the electron beam from the diaphragm center and the shift due to the local fields should be substituted for r. The maximum microscope sensitivity to the magnetic fields on the object will occur under such an initial beam shift when the derivative with respect to the expression (30) is maximum. After a simple calculation of this magnitude, one can obtain that under the maximum sensitivity the relative variation of the electron current to the screen is given by A1 I
= x / ~ " S = Vx
(33)
An experimental example of magnetic field observation by restriction of the electron beam is illustrated in Figure 13. Here the image of magnetic domains on the prismatic face of a Co single crystal was obtained by the above-mentioned method. In the case when the object surface coincides exactly by the prismatic face along which the magnetization vector within the domain is directed, no magnetic stray fields can emerge from the surface. However, if the surface forms a small angle with this direction, part of the magnetic flux starts to emerge from the surface due to the high magnetic anisotropy of Co. The magnitude of this stray field depends on this angle. Here the magnetic field on the surface is rather weak, its magnitude reaches 0.04 T. Domains on this surface look like elongated stripes and wedges. This
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 227 photograph was obtained not by photoelectron emission, but due to the secondary electron emission following ion bombardment. However, these types of electron emission are rather similar from the point of view of the nature of the magnetic contrast. In the crossover plane of the microscope immersion lens, there was an aperture diaphragm that could be displaced to obtain the maximum image contrast [Sedov et al., 1962]. The same method of observation of the local magnetic fields on the surface of an iron-neodymium-boron single crystal (NdzFe14B) by P E E M was also used by Mundschau et al. [1996, 1998]. However, as there was no aperture diaphragm in the microscope, restriction of the beam at the edge of one of the lenses of the electron-optical system was applied in these works. To do this, deflection of the electron beam by an external transverse uniform magnetic field was applied. The quantitative theory given by Dyukov et al. [1991] and Sedov [1970] enables the distribution function of the magnetic field on the object surface to be restored from the images thus obtained. It is of interest to compare the sensitivity to the magnetic fields in the case of restriction of the electron beam and in the case of absence of restrictions described in Section II. For this purpose, the dependence of the electron current density at the microscope screen was calculated for a domain boundary of the first type from Equation (33). The plot of this dependence is demonstrated in Figure 14. Its comparison with Figure 1 shows the different character of the domain boundary imaging under restriction of the electron beams and without it. This calculation was performed for magnitudes close to the real ones: B o = 0.5 T, a - 10-8 m, Vv = 1 V. It is assumed that restriction of the beam is facilitated by an aperture diaphragm placed
J~0
//
1.0 0.5
I
-6
-4
-2
0
2
4
x/a
FIGURE 14. Current density as a function of coordinate in the image calculated for the first type boundary under restriction of the electron beam by the aperture diaphragm.
228
S. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE
in the crossover section in such a way that the image contrast is maximum. It can be seen that the boundary image appears as a brightness jump instead of a dark line. Here the image contrast value was not increased, that is, the contrast depth proves to be significant. Similar results are also obtained for other types of domain boundaries.
IV. MAGNETIC DOMAIN IMAGING IN X - P E E M USING MAGNETIC X-RAY CIRCULAR DICHROISM
A. The Principle of Element-Selective Magnetic Imaging In a PEEM chemical microspectroscopy can be performed exploiting X-ray absorption. The characteristic edge-structure of X-ray absorption features (XANES: X-ray absorption near-edge structure) shows up in the electron emission yield as well. Hence, the intensity of the selected microarea in an image directly reflects the X-ray absorption spectrum if the photon energy is scanned. This technique requires tuneable radiation in the soft X-ray range, for example from a synchrotron source. Pioneering work has been published by Tonner and Harp [1988]. An example of such a chemical spot analysis is shown in Figure 15 [-Swiech et al., 1997]. A microstructured permalloy film (squares of 20 x 20 #m 2) on a silicon wafer has been investigated using local X-ray absorption spectroscopy (XAS). In this mode of operation areas of interest are defined in the image (Figure 15(a)). Area selection is facilitated either electronically by setting regions in the software of the CCD camera or mechanically by choosing the variable iris aperture of the PEEM to the desired size after centering the area of interest. Then, the energy of the exciting radiation is swept in the region of the relevant absorption edges. The electron intensity corresponding to the defined microspot is plotted versus the photon energy (Figure 15(b)). In this case, the XAS-microanalysis of a permalloy square reflects the element distribution of FelgNi81 (lower curve), whereas the analysis of the Si-bars shows the photoyield of Si, which is unstructured in this area (upper curve). Close inspection of the region of the NiL2,3-edges reveals traces of Ni diffusion onto the Si. Given the present conditions at the Berlin storage ring BESSY I (monochromator PM 3 at a bending-magnet beamline), spot sizes down to 500 x 500 nm 2 were possible. At synchrotron-radiation sources of the third generation like BESSY II substantial improvements of microspot resolution will be possible due to the much higher brilliance of undulator beam lines. The magnetic X-ray circular dichroism (MXCD) in the total photoyield at the L2, 3 absorption edges can be utilized for magnetic microspectroscopy.
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 229
FIGURE 15. Chemical microspectroscopy of a permalloy square (20 x 20 ktm 2) array on silicon (a) reflects the element distribution of Fe19Nisl (b) [Swiech et al., 1997].
The dichroic signal arises from the fact that the X-ray absorption cross section at inner-shell absorption edges of aligned magnetic atoms depends on the relative orientation of the photon spin (helicity) and the local magnetization direction. Figure 16(a) shows an example taken at a small spot on one of the permalloy squares of Figure 15. Depending on photon helicity, the spin-orbit split Fe-edges appear in a different intensity I (Figure 16(a), solid and dotted curves). The sign of the dichroic signal AI is reversed at the two spin-orbit split lines of 3d transition metals (Figure 16(b)).
S. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE
230
/,
arb.units
a)
290 -
270 F 250 "'
I
I
I'
,
I
A/, arb.units
b)
5
700
710
720
hv, e V
FIGURE 16. Magnetic microspectroscopy utilizing MXCD (a). The dichroic signal AI (b) arises from the fact that the X-ray absorption cross section at inner-shell absorption edges of aligned magnetic atoms depends on the relative orientation of the photon spin (helicity) and the local magnetization direction. Note that the sign of the dichroic signal is reversed at the spin-orbit split L2.3-shells of 3d transition metals [Swiech et al., 1998].
Figures 16(a) and (b) suggest that high-contrast and laterally resolved magnetic domain patterns can be obtained either by subtracting images acquired at L 2 and L 3 peaks from each other or by subtracting images taken with different photon helicities. This utilization of M X C D in a parallelimaging technique is superior to time-consuming electron spin polarization analysis in a scanning electron microscope for direct imaging of magnetic domains. Schlitz e t al. [1987] have shown by the example of the FeK-edges that absorption of circularly polarized X-rays depends on the magnetization state of the sample. For a fixed photon helicity (left or right circularly polarized) a characteristic change of the absorption spectra has been observed when the magnetization vector M was switched from parallel to
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 231
FIGURE 17. Origin of the magnetic circular dichroism in the electron yield in a simplified single-electron model. The dichroism asymmetry arises in the initial photoexcitation step from the core level into the unoccupied density of states just above E F. The optical spin orientation due to the circularly polarized light leads to different transition probabilities into the majority and minority part of the band structure (different thickness of arrows). The core hole is filled via an Auger decay.
antiparallel orientation with respect to the direction of p h o t o n incidence q or, more precisely, p h o t o n spin s t. This M X C D can be considered as the high energy analog of the magnetooptical Kerr-effect. Both are based on the simultaneous action of spin-orbit coupling and exchange interaction in the electronic states being involved in the optical excitation. The fact that M X C D arises at the X-ray absorption edges provides an outstanding advantage for the element selective investigation of magnetic phenomena. The MXCD-signal occurs in the X-ray absorption and electron yield signal, the latter containing more surface-specific information on magnetism. This effect is well suited for a combination with the P E E M technique for imaging magnetic domains. The M X C D in the initial absorption signal is transferred to the emitted electrons in a 2-step-process; see Figure 17. First,
232
S. A. N E P I J K O , N. N. SEDOV AND G. S C H O N H E N S E
the optical excitation creates a core hole, in our example, in the 2p3/z-shell. Owing to optical spin orientation the excited electrons are spinpolarized [Meier and Zakharchenya, 1984]. Close to the absorption edge, the final state of the initial photoexcitation lies in the region of the unoccupied d-band above the Fermi energy E F. Since there is a higher unoccupied density of states in the minority spin channel (right density of states in Figure 17), primary electrons with this spin orientation are favored, whereas the majority spin direction finds only a small part of unoccupied band structure. This is the origin of the MXCD asymmetry in the initial absorption step. The resulting different absorption cross sections for opposite magnetization directions is equivalent to a different probability of the creation of a core hole. The spin quantization axis for the density of states is defined by - M ; for the optical spin orientation it is the photon spin s t. The core hole decays with a final lifetime either through fluorescence or via an Auger process. In the latter case the magnetic dichroism in the absorption channel is directly transferred to the Auger electron yield. For parallel and antiparallel configuration of M and s t one obtains different intensities of the Auger transitions. This magnetic circular dichroism in the photon-induced Auger-electron emission can be utilized for energy selective imaging of magnetic domains (i.e., electron-energy resolved) as shown by Schneider et al. [1993]. St6hr et al. [1993] performed the pioneering experiments of MXCD spectromicroscopy using a PEEM. In a standard PEEM (without energy filter) it is not possible to use the Auger-electron signal as magnetic contrast because the total electron energy distribution is strongly dominated by the secondary electrons. On their way to the sample surface the characteristic Auger electrons experience inelasting scattering events and thus produce a cascade of secondary electrons. In a good approximation, the intensity of these secondary electrons is proportional to the number of the initially excited Auger electrons. In this way, the MXCD-signal is finally transferred via the intermediate step of the Augerelectron emission to the low-energy secondary electrons. Except for the MXCD asymmetry, these electrons carry no direct information about the specific electronic transition in the sample. However, element selectivity is ensured by the initial excitation at a characteristic absorption edge. Since the cathode lens acts as an efficient low-pass filter, these secondary electrons in the region between threshold and approximately 3-5eV are used for forming the image in the PEEM. The exploitation of the MXCD-signal in the secondary electron yield as contrast mechanism allows a direct imaging of the domain structure of a selected element in a ferromagnetic sample. The parallel image acquisition in a PEEM and the combination of high magnetic contrast and high intensity of the secondary electrons facilitates very short exposure times. The
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 233 method allows an aberration-free imaging of magnetic microstructures. Unlike in magnetic force microscopy the ferromagnetic domain structure is not influenced by the P E E M technique if an electrostatic objective lens (in combination with an effective mu-metal shielding) ensures the sample region being free of magnetic stray fields. This method was utilized by several groups for magnetic imaging during the last years [Bauer et al., 1997; Fecher et al., 1999; Kuch et al., 1998; Schneider, 1997; Schneider et al., 1997; Spanke et al., 1998; St6hr et al., 1993; Swiech et al., 1998; Tonner et al., 1994].
B. M i c r o p a t t e r n e d S t r u c t u r e s a n d D o m a i n Walls
Figure 18 shows domain patterns of periodic microstructures of permalloy (a), a Co/Pt-multilayer (b) and an epitaxial Co film (c) [Swiech et al., 1997; Schneider, 1997; Schneider et al., 1997; Kuch et al., 1998]. The first two samples were evaporated on a Si surface covered by native oxide, whereas the third one was grown on a Cu(100) single crystal surface. The size of the squares was 20 x 20/~m 2 in (a) and (b) and 8 x 8 #m 2 in (c). All measurements were made at room temperature. In the case of the permalloy structure (Figure 18(a) the domain pattern is made visible "in the light of" the FeLz,3-edges [Swiech et al., 1997]. The image represents the difference of two images taken at the FeL 3 and L2-edge. Since the MXCD-signal changes its sign when switching from the L 3 to the Lz-edge (cf. Figure 16(b)), the magnetic information is enhanced effectively by suppressing nonmagnetic contrast contributions. Most permalloy
FIGURE 18. Magnetic domain structure in square arrays of thin films of permalloy (a), a Co/Pt-multilayer (b) and Co on Cu(100) (c). The sizes of the squares are 20 x 20 # m 2 (a and b) and 8 x 8 # m 2 (c). The different behavior of the magnetic anisotropy causes different domain patterns [Swiech et al., 1997; Schneider, 1997; Schneider et al., 1997; Kuch et al., 1998].
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squares exhibit a very regular domain structure: four triangles, two of which appear in intermediate gray level, whereas the other two appear darker and brighter. This distribution of the contrast can be understood by the angle dependence of the MXCD-signal proportional to the scalar product M. s~. Thus for reasons of symmetry the magnetic contrast vanishes for an orthogonal arrangement of M and sy. The intermediate gray level indicates such domains, whereas for the bright and dark domains M is parallel or antiparallel with respect to s~ except for a factor of cos25 ~ due to the direction of photon impact. This consideration allows reconstruction of the magnetization distribution within the squares. It has the shape of a typical flux-closure structure, that is, each square tends to minimize its outer magnetic stray field. This configuration represents the simplest case of a domain pattern and is obviously favored by the vanishing magneto-crystalline anisotropy of permalloy. However, this ideal case can be perturbed by defects on the surface, as visible for the square in the center of the image. An interesting aspect for future experiments is a possible magnetic coupling between the squares and its influence on the domain pattern. A material with a strong magneto-crystalline anisotropy, like the Co/Ptmultilayer, shows a completely different behavior, see Figure 18(b) [Schneider, 1997; Schneider et al., 1997]. In this case the magnetic information has been obtained at the CoL3-edge by changing the photon helicity. Again, the magnetic signal is reversed and in the difference image only the magnetic contrast appears. The resulting domain structure is much more complex as in the preceding case and varies in detail between the different squares. The feather-like features indicate locally varying easy directions of magnetization. This result is consistent with the polycrystalline character of the Co/Pt-layers. It represents a first example for the investigation of "buried layers." The topmost layer of the Co/Pt-stack consists of approximately 3 nm Pt. The element selectivity and information depth of the method allows investigation of the magnetic signal of the Co layers through the Pt top layer. The third example (Figure 18(c)) shows another different behavior [Kuch et al., 1998]. In this case an epitaxial Co film of 15 monolayers (ML) thickness grown on a clean Cu(100) substrate has been investigated. The domain structure of the film has been observed at the CoL 3 edge in the "as-grown" magnetic state. The [100J-direction of the substrate was oriented parallel to the plane of incidence (indicated by the arrow). This leads to four equivalent easy axes of the in-plane magnetization of the Co film. A spontaneous magnetization of the Co squares along the four (110) directions would thus result in equal projections of either two of these easy axes along the direction of photon incidence. In turn, this would lead to the observation of only two different asymmetries in the experiment. The
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experimental result of only two gray levels is thus compatible with the Co structures being magnetized along one of these fourfold crystallographic axes. With few exceptions only (e.g., the bottom-left square), the Co squares appear in single-domain states. The simple angular dependence M . s t of the MXCD signal can also be exploited to selectively image domain walls. This experiment is based on the following consideration: In a domain wall the direction of the magnetization vector varies continuously across the interface of two domains. In a Bloch wall occurring in the bulk, M rotates about an axis perpendicular to the plane of the wall. At a surface, however, such a wall would generate a magnetization component perpendicular to the surface. This state is energetically unfavorable and is avoided by the formation of a N6el-like wall. In this case M rotates about an axis perpendicular to the surface, that is, M remains in the plane of the surface. This N6el-termination of Bloch walls at the surface is well-known from scanning electron microscopy with spin polarization analysis (SEMPA) [Oepen and Kirschner, 1989]. In the region of the N6el-wall a magnetization component occurs, which lies in the surface and is perpendicular to the domain magnetization direction. In a system of fourfold symmetry like Fe(100) it has the consequence that for the geometries of M]] s t or M_l_s t one can selectively image either the domains or the domain walls, respectively. An image of the domain walls in the surface of a Fe(100) single crystal (whisker) is shown in Figure 19 [Schneider et al., 1997]. In the major part of the image no magnetic domain contrast is visible because M is orthogonal to the plane of photon incidence. However, two pronounced zig-zag lines are observed. The contrast of each line changes from bright to dark in the positions indicated by the circles. The magnetization direction of the domains are indicated by the black arrows. In principle, this magnetization distribution is energetically unfavorable and becomes stabilized by the bulk. It occurs when the magnetization deep in the bulk is oriented perpendicular to the surface. In order to reduce the magnetostatic energy, flux-closure domains are formed in the surface region, whose magnetization vectors lie in the surface plane. Consequently, domain walls in Figure 19 are no simple 180~ but rather so-called "V-lines" being indicative of 90~ coming from the bulk and meeting at the surface [Chikazumi, 1994]. Their width is approximately 500 nm. The resulting angle at the kinks of the curve in the image is 106 ~, which agrees very well with the value known for V-lines in Fe(100). A change of contrast observed along the line results from the fact that for N6el-like terminations of the domain walls at the surface two different senses of rotation of the magnetization are possible. Hence, the rotation sense along the wall can flip from clockwise to counterclockwise (see circles). This image has been taken at the European Synchrotron
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FIGURE 19. Domain walls in the surface of an Fe(100) single crystal. Arrows denote the local orientations of the magnetization and circles mark the change of the sense of rotation of the magnetization in a domain wall, which leads to a contrast reversal [Schneider et al., 1997].
Radiation Facility (ESRF) with exposure times of about 5 minutes for each helicity, thanks to the high brightness of the undulator beamline ID 12 B. In the future, the detailed investigation of such structures and discontinuities will be one of the major challenges of magnetic photoemission microscopy.
C. Exchange-Coupled Systems and Probing Depth Exchange-coupling in magnetic heterosystems (multilayers, sandwich structures) is one of the highly interesting issues in thin-film-magnetism in view of its potential for applications. Such heterosystems are the basis for GMR sensors (giant magnetic-resistance) [Grfinberg, 1986] and basic building elements (spin valves, magnetic tunnel junctions) for devices of spin electronics like the M-RAM (magnetic random access memory) [Tang et al., 1995]. The potential to view "buried layers" through top layers of different constituents using the element-resolved imaging technique is one of the striking advantages of MXCD-PEEM. Except for the trivial case of a nonmagnetic top layer none of the other imaging techniques is capable to reveal the magnetic structure of a buried layer in an exchange-coupled structure. In the following we will present an example in order to illustrate this powerful tool.
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FIGURE 20. Investigation of the magnetic information depth in a Co/Cr/Fe(100) sandwich. The thickness of the Cr-wedge increases from the left to the right as indicated in the top panel. The element resolved domain images for Co, Cr, and Fe have been taken at the corresponding L2, 3 absorption edges. The field of view is 500 #m horizontally [Schneider, 1997; Schneider et al., 1997].
Figure 20 shows the sandwich structure of a Co film being separated from an Fe substrate by a Cr-wedge [Schneider, 1997; Schneider et al., 1997]. The sample was prepared in the following way: The Cr-wedge with increasing thickness from 0-10 ML was grown on a Fe(100) whisker (single crystal) surface at about room temperature. Finally the whole structure was covered by a 5 ML Co film. The sandwich structure is schematically illustrated in the top panel of Figure 20. The system was imaged element selectively at the L3-edges of Co, Cr, and Fe. The typical domain structure of the Fe whisker, two regions of opposite magnetization, is clearly visible in these single images without magnetic contrast enhancement. Bright areas correspond to a magnetization direction opposite to the direction of photon incidence and dark areas to magnetization along the photon incidence direction. The simple domain configuration often encountered in Fe whiskers is very convenient
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for imaging magnetic coupling phenomena in wedge-shaped overlayers [Schneider et al., 1996; Unguris et al., 1997]. The decrease of brightness and contrast in the Fe image (bottom) from left to right is a consequence of the degradation of the signal due to the increasing thickness of the top layers. Nevertheless, the magnetic structure is visible through 10 ML Cr plus 5 ML Co, that is, a total thickness of almost 5 nm. St6hr et al. [1998] have found that the MXCD signal can still be detected through a 10nm thick layer of Rh or Ag. Siegmann [1992], Sch6nhense and Siegmann [1993] have shown that the escape depth in transition metals is largely determined by scattering processes from filled to empty states and is, therefore, inversely proportional to the number of d-holes. According to this rule, the magnetic probing depth strongly depends on the material of the top layer. It is highest for materials with completely filled d bands like Cu, Ag, or Au. For nonmetallic top layers there is not much information at present. Ade et al. [1998] have successfully detected Ti dots buried underneath as much as 40 nm of A1N grown by chemical vapor deposition (CVD) in PEEM with ultraviolet (UV) excitation at hv = 5-6 eV. Since A1N has a large band gap of ~6.2 eV and a negative electron affinity, however, this may represent a special case. The Cr-selective image shows the onset (dashed line) and increase of the Cr-wedge. A domain structure in Cr is not visible on this length scale (field of view 500/~m horizontally), even with magnetic contrast enhancement. Finally, the Co-selective image shows the domain structure in the Co top layer. Interestingly, in this case the Cr layer is coupled ferromagnetically to the substrate all along the Cr-wedge. This behavior depends crucially on the preparation conditions as has been shown by Kuch et al. [1998]. For 500 K substrate temperature during evaporation the coupling character of this system has changed. For a Cr thickness above about 2 ML the magnetic contrast in the Co image is reversed with respect to the Fe image. At these Cr thicknesses, Co displays an antiferromagnetic coupling to the Fe substrate. In addition the Cr-wedge also shows an MXCD signal. A significant influence of the amount of interface intermixing on the occurrence of the antiferromagnetic interlayer coupling [Freyss et al., 1997] may play a role. From these results it becomes clear that elemental selectivity is an absolutely necessary prerequisite for studying the coupling behavior of layered magnetic structures. These images are obtained in the so-called "survey mode" with the objective lens being operated in the mode of an electrostatic triode and being characterized by a low voltage of extractor electrode [Sch6nhense]. In this way, a field of view of more than 0.7mm is possible with a corresponding resolution of about 1/~m.
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 239 V. MAGNETIC DOMAIN IMAGING IN U V - P E E M KERR-EFFECT-LIKE CONTRAST
USING A
A. Magnetooptical Kerr-Effect and Its Manifestation in Threshold PEEM Optical Kerr-microscopy is up to now the most common technique for magnetic domain imaging [Hubert and Sch/ifer, 1998]. The magnetooptical Kerr-effect results from a dependence of the dielectric tensor ~ from the magnetization of the sample. It causes a small rotation of the electric vector upon reflection on the sample surface as well as a magnetization-dependent intensity modulation of the reflected light beam in a special geometry. Depending on the relative orientation of magnetization and plane of incidence one can distinguish between longitudinal (M [[ plane of incidence) and transversal Kerr-effect (M_t_ plane of incidence) [Erskine and Stern, 1973]. Kerr microscopy employs light in the visible or near UV-range and is diffraction limited. By using an immersion objective lens and an optimized dielectric coating of the magnetic sample surface, resolutions down to about 300nm can be achieved in favorable cases. In contrast, photoemission microscopy is characterized by a base resolution of better than 20 nm. In Section IV we have discussed that imaging of magnetic domain patterns with PEEM is possible using circularly polarized synchrotron radiation and exploiting magnetic circular dichroism. This dichroism effect occurs at characteristic absorption edges and, hence, the approach depends on the access to a tunable light source in the soft X-ray range. In threshold photoemission (UV-PEEM) only electrons in a narrow energy window of typically less than 1 eV can contribute to the image. The amount of inelastically scattered secondary electrons is small and the entire spectrum is dominated by direct photoelectrons. In addition, due to their low kinetic energy the excited electrons are strongly refracted upon leaving the crystal. Therefore, the image is formed by electrons having only a small wave-vector component ktt to the surface, that is, close to the normal emission condition. Threshold photoemission has been theoretically treated e.g. by Maran [1970], and Sass [1975]. We will explain the origin of the magnetic contrast by using a simple model known from the description of the transversal magnetooptical Kerr-effect; see Figure 21. In a magnetized sample the dielectric response on the external electric field vector has two m a i n contributions. The first is the refraction at the solid-vacuum interface leading to an orientation of the displacement vector D t after transmission more parallel to the surface as compared with the electric vector Ei of the incident light wave; see Figure 21.
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s. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE
n
Q M
gi
O, +
| M
'1'n
_
I\
gr
O,
FIGURE 21. Origin of the Kerr-effect-like magnetic contrast in threshold photoemission. The phenomenon arises in the geometry used to measure the transversal magnetooptical Kerr-effect and has the same origin. Explanation--see text.
Second, the Lorentz force on the quasi-free metal electrons leads to a small additional Kerr-rotation of the displacement vector D,+k, depending on the local magnetization vector M as illustrated in Figure 21. This rotation is the origin of the transversal magnetooptical Kerr-effect, that is, an intensity modulation in the reflected photon beam, characterized by Er. It also gives rise to the magnetic contrast discussed in this section. Since the photoelectrons contributing to the image have momenta in a small, solidangle interval around the surface normal, the projection of Dt+k onto the surface normal n is a measure of the intensity of the emitted electron signal. In the simple picture of quasi-free electrons, that is, neglecting all bandstructure effects, the emission intensity I is thus proportional to the square of the scalar product I oc (Dr +k" n) 2. In Figure 21 we have assumed a magnetization vector perpendicular to the drawing plane and pointing downwards in the right-hand domain and upwards in the left-hand domain. The resulting rotation of the displacement vector is thus counterclockwise or clockwise, respectively. In the case of the counterclockwise rotation the projection of D t +k onto n is larger, thus giving rise to a higher photoelectron intensity in threshold emission than in the opposite case. The corresponding domain (right) will therefore appear brighter than the other one. Such magnetization-dependent intensity differences in photoemission with linearly polarized light are usually termed
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241
linear magnetic dichroism (LMD). They have been observed in spectroscopy experiments for various magnetic materials mostly in the X-ray range [Hillebrecht et al., 1995; Roth et al., 1993] and vacuum ultraviolet (VUV) [Rampe et al., 1998-1, but also in threshold photoemission [Fanelsa, 1996]. The schematic principle of Figure 21 illustrates that the orientation of the magnetization vector on the right-hand side leads to an enhanced LMD signal and also to an enhanced reflected intensity in the Kerr-signal. More details on this analogy are discussed by Marx et al. The simple picture of quasi-free electrons being based essentially on the Drude-theory extended for the presence of a sample magnetization is particularly useful for polycrystalline materials, where no k-resolution is given. For the case of single-crystal surfaces the microscope detects electrons around a well-defined direction in k-space. In this case the so-called magnetic linear dichroism in the angular distribution (MLDAD) must be considered. Henk et al. [1994] have performed an ab-initio calculation for the Ni(110) surface at hv = 5.1 eV photon energy, which indeed predicts the existence of a magnetic dichroism in particular for sp-polarized light, that is, with the electric vector being rotated by 45 ~ in comparison to the situation depicted in Figure 21. This type of dichroism occurs already without inclusion of the surface optical response. A detailed analysis of the results for excitation with sp- and circularly polarized light reveals strong similarities in the electronic origin of the associated magneto-dichroic phenomena (see also Marx et al. and Sch6nhense).
B. First Detection o f the Kerr-Effect-Like Contrast in U V - P E E M
Marx et al. have performed the first experiments to detect the novel magnetic contrast mechanism in threshold photoemission. The experimental geometry corresponds to a typical arrangement for detecting the transversal Kerr-effect with an angle of photon incidence of 75 ~ with respect to the surface normal. The photon beam from a Xe-Hg high pressure arc lamp (100 W) passed through a linear polarizer (Glan-Thompson prism) and was focused onto the sample surface. By rotating the prism the linear polarization could be changed continuously between s- and p-polarized states. A polycrystalline Fe film with a thickness of 100 nm has been deposited by means of ultrahigh vacuum (UHV) evaporation on a Si wafer with native oxide layer. The sample holder allowed the in-situ application of an external magnetic field. The coercive field of the Fe film was determined to be about #oH = 4 mT. The work function of the film was 9 = 4.8 eV, so that the energy width of the photoelectron distribution was approximately AE ~< 0.5 eV.
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In Section IV it has been mentioned that the magnetic contrast can be enhanced by subtracting a suitable "background-image." In threshold photoemission this background image can be taken with either a completely demagnetized or a fully magnetized sample, a technique known from Kerr-microscopy [Hubert and Sch~ifer, 1998]. In the resulting difference image, other contrast contributions such as work function contrast, topographical contrast, impurities, and so on are largely eliminated. Figure 22(a) shows an example of the resulting magnetic contrast between two oppositely magnetized domains. In this case a background image of the sample being in a single-domain state has been subtracted. The asymmetry value extracted from regions of opposite magnetization is A = (0.37 _ 0.10)%. Quantitatively, this asymmetry agrees well with the corresponding value for the magnetooptical Kerr-effect measured for Fe [Dove, 1963; Katayama et al., 1998]. As compared to the demanding experimental requirements of circularly polarized tunable radiation in the soft X-ray range, the new approach is extremely simple. It works with a standard laboratory UV-source. If we take into account that unpolarized light contains a 50% contribution of ppolarization (which could be enhanced inside the material due to metal optics) the mechanism described in Figure 21 should survive. Indeed, Marx et al. were able to image the magnetic domains in the polycrystalline Fe film using unpolarized light; see Figure 22(b). In this case the magnetic asymmetry is A = 0.18%, that is, about half of the value obtained for linearly polarized light. The field of view is about 120 • 120/~m 2 at a resolution of the CCD camera of 1024 • 1024 pixel. The total exposure time was 2 minutes. There is no need to change the magnetization of the sample for taking a background image. In principle, a second UV-lamp placed in the azimuth opposite to the first one allows taking the second image with the magnetic contrast being reversed. Although the first results have been measured for thin Fe films, the phenomenon has the same general physical origin as the magnetooptical Kerr-effect. It will thus not be restricted to a specific class of materials. Due to its potential for high lateral resolution and the efficient parallel image acquisition, the new method is highly attractive for applications. It is a relatively simple laboratory method and does not require special light sources such as synchrotron radiation.
VI. CONCLUSIONS The technique of PEEM allows observation of the domain boundaries on ferromagnetic surfaces arising from magnetic stray fields at definite condi-
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 243
FIGURE 22. Domain patterns of a polycrystalline Fe film obtained in threshold photoemission with linearly polarized (a) and unpolarized UV-light (b). The images were taken using a Hg-Xe laboratory UV-source in the geometry of Figure 21 [Marx et al.].
tions. In most cases a very weak contrast due to the Lorentz force is formed. Nevertheless, it can be measured by the use of digital methods of signal registration and data processing. From the form of the observed signal, conclusions about the shape of the stray field of the domain boundary can be drawn if there is additional information. For example, one can determine
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S. A. NEPIJKO, N. N. SEDOV AND G. S C H O N H E N S E
whether the turn of the magnetization vector within the boundary is of the right- or left-handed screw type or what the maximum value of the stray field is and what magnitude has the half-width of its distribution. The only case of high-image contrast is observed for ferromagnets with strong anisotropy. In this case not only the stray fields of the boundaries, but also the fields of the domains themselves can emerge from the object surface. However, in the case of strong fields an image distortion appears and the quantitative theory of image formation should be used for decoding this distortion. If the electron beam is restricted by either an aperture diaphragm or a knife-edge being located in the crossover plane (back focal plane of the objective lens), the sensitivity of PEEM to the magnetic fields is strongly enhanced. This permits comparatively weak stray fields of the domain boundaries to be observed. The character of the boundary images changes as well, but it can be analyzed by the theory of image formation, and the value of the stray fields can be derived. However, it is necessary to use a special microscope construction for this method of observation. An aperture diaphragm or a knife-edge should be placed in the crossover plane of the immersion lens. It is desirable to have the possibility of displacement of this element in the direction normal to the microscope optical axis in order to obtain the optimum image contrast. In the case of excitation using circularly polarized radiation it is possible to exploit MXCD for contrast formation. This contrast arises as a consequence of optical spin orientation (mediated by the spin-orbit interaction) and a spin asymmetry in the unoccupied part of the electronic band structure (due to exchange splitting). The effect occurs at the characteristic X-ray absorption edges, therefore this method is element specific. In addition, layered and buried magnetic structures are accessible. Materials containing different constituents with different magnetic behavior (e.g., temperature dependence) or the magnetization of films in the submonolayer coverage range are further fields of application. Such experiments require tuneable and circularly polarized X-rays and hence can be performed at synchrotron radiation sources only. A very recent achievement is a magnetic imaging mode of a PEEM in threshold photoemission, which makes use of a Kerr-effect-like rotation of the displacement vector inside a magnetic material. It occurs in the geometry of the transverse Kerr-effect, that is, with the magnetization perpendicular to the plane of photon incidence. Depending on the local direction of magnetization this rotation results either in an enhancement or in a reduction of the component of the displacement vector perpendicular to the surface. In turn, the corresponding photoelectron intensity depends on the local magnetization direction.
MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES
245
If we compare the information contents of the Lorentz-force contrast (Sections II and III) and the Kerr-effect-like contrast (Section V), both being accessible with simple UV-sources, there are important differences. The former contrast is based on the existence of a magnetic stray field close to the sample surface. Therefore, the method is well suited for perpendicularly magnetized structures and for the observation of domain walls. A homogenous in-plane magnetization of a thin film will generally not lead to a contrast. A thin-film structure (like a square) being uniformly magnetized will essentially show up in the regions of the magnetic poles. In this respect the technique bears much resemblance to the magnetic force microscopy. In contrast, optical Kerr-microscopy as well as its counterpart in photoemission microscopy yields a domain contrast, that is, flat areas of unbroken gray level. In this case the phenomenon arises inside of the material. A thin-film structure being uniformly magnetized will therefore appear in a uniform gray level. The resolution of a PEEM typically can be driven to 20 nm. This and the fast parallel image acquisition makes the technique highly attractive for applications in the modern fields of magnetic storage devices or spin electronics development. In principle, the resolution can be further enhanced by correction elements, but theory as well as computer simulation [Nepijko et al.] has revealed that a surface corrugation as well as residual electric fields (i.e., due to work function differences) or magnetic stray fields (as discussed in Sections II and III) set a practical limit to the attainable ultimate resolution.
ACKNOWLEDGMENTS
The authors thank P. Hawkes for helpful discussion as a result of which the topic of the present chapter was chosen. The work was in part carried out during a research stay of S.A.N. at the Fritz-Haber-Institut der MaxPlanck-Gesellschaft, Berlin. G.S. would like to thank his coworkers Ch. Ziethen, G. K. L. Marx, O. Schmidt, and G. Fecher, as well as all collaborators involved in the experiments of Section IV, in particular W. Kuch, R. Fr6mter, and J. Kirschner (Max-Planck-Institut fiir Mikrostrukturphysik, Halle), C. M. Schneider (Institut fiir Festk/Srper- und Werkstofforschung, Dresden) and W. Swiech (University of Urbana, Illinois). These experiments have been funded by BMBF (05621 UMA7, 05644 UM, and 05644 EFA), by D F G through Sonderforschungsbereich 262 and by Materialwissenschaftliches Forschungszentrum Mainz.
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S. A. NEPIJKO, N. N. SEDOV AND G. SCHONHENSE REFERENCES
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MEASUREMENT OF MAGNETIC FIELDS AND DOMAIN STRUCTURES 247 Nepijko, S. A., Sedov, N. N., SchiSnhense, G., Escher, M., Bao, X., and Huang, W. Resolution deterioration in emission electron microscopy due to object roughness (to be published). Oepen, H. P., and Kirschner, J. (1989). Magnetization distribution of 180 ~ domain walls at Fe(100) single-crystal surfaces. Phys. Rev. Lett. 62, 819-822. Rampe, A., Giintherodt, G., Hartmann, D., Henk, J., Scheunemann, T., and Feder, R. (1998). Magnetic linear dichroism in valence-band photoemission: Experimental and theoretical study of Fe(ll0). Phys. Rev. B 57, 14370-14380. Recknagel, A. (1941). Theorie des elektrischen Elektronenmikroskops ffir Selbststrahler. Z. Phys., 117, 689-708. Roth, Ch., Rose, H. B., Hillebrecht, F. U., and Kisker, E. (1993). Magnetic linear dichroism in soft X-ray core level photoemission from iron. Solid State Commun. 86, 647-650. Sass, J. K. (1975). Evidence for an anisotropic volume photoelectric effect in polycrystalline nearly free electron metals. Surf Sci. 51, 199-212. Sedov, N. N. (1970). Th6orie quantitative des syst6mes en microscopie 61ectronique 5. balayage, /t miroir et/t 6mission. J. Microsc. 9, 1-26. Scheunemann, T., Halilov, S. V., Henk, J., and Feder, R. (1994). Spin reversal and circular dichroism in valence-band photoemission from 3d-ferromagnets. Solid State Commun. 91, 487-490. Schneider, C. M. (1997). Soft X-ray photoemission electron microscopy as an element-specific probe of magnetic microstructures. J. Magn. Magn. Mater. 175, 160-176. Schneider, C. M., Fr6mter, R., Ziethen, Ch., Swiech, W., Brookes, N. B., Sch6nhense, G. and Kirschner, J. (1997). Magnetic domain imaging with the photoemission microscope. Mat. Res. Soc. Symp. Proc. 475, 381-392. Schneider, C. M., Holldack, K., Kinzler, M., Grunze, M., Oepen, H. P., Sch/ifers, F., Petersen, H., Meinel, K., and Kirschner, J. (1993). Magnetic spectroscopy from Fe(100). Appl. Phys. Lett. 63, 2432-2434. Schneider, C. M., Meinel, K., Kirschner, J., Neuber, M., Wilde, C., Grunze, M., Holldak, K., Celinski, Z., and Baudelet, F. J. (1996). Element specific imaging of magnetic domains in multicomponent thin film system. J. Magn. Magn. Mater. 162, 7-20. Sch6nhense, G. (1999). Imaging of magnetic structures by photoemission electron microscopy. J. Phys.: Condens. Matter. 11, 9517-9547. Sch6nhense, G., and Siegmann, H. C. (1993). Transition of electrons through ferromagnetic material and applications to detection of electron spin polarization. Ann. Phys. 2, 465-474. Schiitz, G., Wagner, W., Wilhelm, W., Kienle, P., Zeller, R., Frahm, R., and Materlik, G. (1987). Absorption of circularly polarized X rays in iron. Phys. Rev. Lett. 58, 737-740. Siegmann, H. C. (1992). Surface and 2D magnetism. J. Phys.: Condens. Matter. 4, 8395-8434. Spanke, D., Solinus, V., Knabben, D., Hillebrecht, F. U., Ciccacci, F., Gregoratti, L., Marsi, M. (1998). Evidence for in-plane antiferromagnetic domains in ultrathin NiO films. Phys. Rev. B 58, 5201-5204. Spivak, G. V., Dombrovskaya, T. N., and Sedov, N. N. (1957). The observation of ferromagnetic domain structure by means of photoelectrons. Soy. Phys. Dokl. 2, 120-123. St6hr, J., Padmore, H. A., Anders, S., Stammler, T., and Scheinfein, M. R. (1998). Principles of X-ray magnetic dichroism spectromicroscopy. Surf Rev. and Lett. 5, 1297-1308. St6hr, J., Wu, Y., Hermsmeier, B. D., Samant, M. G., Harp, G. R., Koranda, S., Dunham, D., and Tonner, B. P. (1993). Electron-specific magnetic microscopy with circularly polarized X-ray. Science 259, 658-661. Swiech, W., Fecher, G. H., Ziethen, Ch., Schmidt, O., Sch6nhense, G., Grzelakowski, K., Schneider, C. M., Fr6mter, R., Oepen, H. P., and Kirschner, J. (1997). Recent progress in photoemission microscopy with emphasis on chemical and magnetic sensitivity. J. Electron Spectrosc. Relat. Phenom. 84, 171-188.
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Swiech, W., Fr6mter, R., Schneider, C. M., Kuch, W., Ziethen, Ch., Schmidt, O., Fecher, G. H., Sch/Snhense, G., and Kirschner, J. (1998). Magnetically resolved and element specific imaging with photoelectrons using an immersion lens column. In: Electron Microscopy. Cancun, Mexico, p. 511-512. Tang, D. D., Wang, P. K., Speriosu, V. S., Le, S., and Kung, K. K. (1995). Spin-valve RAM cell. IEEE Trans. Magn. 31, 3206-3208. Tonner, B. P., Dunham, D., Droubay, T., and Pauli, M. (1997). A photoemission microscope with a hemispherical capacitor energy filter. J. Electron Spectrosc. Relat. Phenom. 84, 211-229. Tonner, B. P., Dunham, D., and Zhang, J. (1994). Imaging magnetic domains with the X-ray dichroism photoemission microscope. Nucl. Instrum. Methods. Phys. Res. A 347, 142-147. Tonner, B. P., and Harp, G. R. (1988). Photoemission microscopy with synchrotron radiation. Rev. Sci. Instrum. 59, 853-858. Unguris, J., Celotta, R. J., and Pierce, D. T. (1997). Determination of the exchange coupling strengths for Fe/Au/Fe. Phys. Rev. Lett. 79, 2734-2737.
ADVANCES IN I M A G I N G AND ELECTRON PHYSICS, VOL. 113
Improved Laser Scanning Fluorescence Microscopy by Multiphoton Excitation N. S. W H I T E 1 A N D R. J. E R R I N G T O N 2 aBio-Rad Biological Microscopy Unit (BMU), Department of Plant Sciences, University of Oxford, South Parks Road, Oxford. UK OX1 3RB 2Department of Medical Biochemistry, University of Wales College of Medicine, Heath Park, Cardiff UK CF14 4XN
I. Introduction . . . . . . . . . A FluorescenceMicroscopy B Laser Scanning Microscopy C. Confocal Microscopy . . . . D Multiphoton LSM . . . . . II Future Prospects . . . . . . . References . . . . . . . . . .
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I. INTRODUCTION
The compound optical microscope has been central to countless advances in biological research Unchanged in principle since the earliest forms of the modern instrument, well over a hundred years ago, refinements have concentrated on increasing performance to near-theoretical limits together with methods to visualize specific optical features of the specimen The growth of cell and molecular biology in recent years has forced the pace of microscopy research to provide high contrast images of increasingly more specific structures and functions in biological specimens One technique is fluorescence microscopy [-Ploem and Tanke, 1987; Wang and Herman, 1996; Rost, 1992], where highly specific chemical probes are used to stain particular components of the sample and imaged by their characteristic emission spectra. The method has advanced rapidly through parallel developments in optical filters and lenses, laser and nonlaser light sources, and highly efficient detectors. A. Fluorescence M i c r o s c o p y
Available in various forms as an attachment or modification for all standard research microscopes, the fluorescence contrast technique relies on the 249 Volume 113 ISBN 0-12-014755-6
ADVANCES IN I M A G I N G AND E L E C T R O N PHYSICS Copyright 9 2000 by Academic Press All rights of reproduction in any form reserved. ISSN 1076-5670/00 $35.00
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,
excited states
"--,,,,
0 \
lk
f_
1-P
i \ I
increasing energy 9
9
_jo
fluorescence emission ground state
FIGURE 1. Jablonski energy diagram for conventional (single-photon) fluorescence excitation. Fluorescent probe molecules are raised to a higher energy singlet state by the absorption of a single photon of illumination light. A small amount of energy is lost as the molecule falls to the lowest energy level possible by dissipation through internal vibration events. A random time after absorption (typically a few ns for common probes) the molecule finally decays back to the low energy ground state by the emission of a single photon of fluorescent light. The difference in energy between the final excited state and the ground state is reflected in a wavelength change (Stokes shift) between absorbed and emitted (longer wavelength) photons.
combination of three highly optimized components: 1. A light source to illuminate the specimen with a well-defined spectrum. 2. A detection system to collect light emitted from the specimen with an equally well-defined spectrum. 3. A fluorescent probe or stain that absorbs all or part of the illuminating light and emits fluorescence in the spectral range of the detection [Dewey, 1991; McGown and Warner, 1994; Haugland, 1995]. Normally, a fluorescent molecule will absorb a single photon of the illuminating light and be excited to a highly energetic state (Figure 1). After a short but random period of time (typically a few ns) some of this energy is lost as the molecule relaxes to a slightly lower energy before the molecule returns to the original (or ground) state by emitting a photon of light. The random delay results in complete loss of phase information from the illumination and the fluorescent light is emitted in all directions. Since some energy is lost before the photon is emitted, the wavelength of fluorescence is longer than that of the exciting light. Fluorescence microscopy is able to deliver the high contrast needed to distinguish the fluorescence light of stained structures (and functions) from a background of unwanted non-
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specific fluorescence and reflected light features by the use of high-specification interference filters.
1. Optical Performance of a Microscope An important requirement in any optical microscope is to be able to distinguish two fine structures in a specimen separated by a small distance within the plane of focus. This is usually described as the in-plane resolving power or resolution and is largely dependent on the objective lens NA and the wavelength (2) of light used to form the image. This last point is most important in the discussion of different microscopes. For a single objective lens, when two extremely small point objects can be just discerned as separate features they are said to be resolved by the Raleigh criteria (see Pawley [1995], Ch. 1) and their centers will be separated by a distance dxy, where d x y "-"
0.612 NA
A related measure (dz) can be derived for point objects to be resolved in the axial (focus) direction:
dz
3.28dxyr/ NA
where r/is the refractive index of the specimen. Clearly in both cases, shorter wavelengths give rise to a smaller resolved separation (i.e., higher resolution) and the ratio of in-plane to axial resolution is always worse than a factor of 3. It might also be noted that the in-plane resolution varies linearly with NA whereas the axial resolution changes with the square of this important objective lens parameter. There are other standard measures of resolution, differing in the contrast that is achieved between the object and the background. Any factor that reduces the image contrast also degrades the resolution attainable.
2. Conventional Fluorescence Microscopy In the conventional fluorescence microscope, the illuminating light is directed over the entire area of the specimen. The objective lens forms a complete image of the fluorescence emitted in all directions from all parts of the specimen in the field of view. This has the important effect of making the in-plane and axial resolution dependent on the wavelength of just the fluorescence light. The spectral characteristics of the exciting light do not determine the resolving power.
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3. Out-@Focus Blur One cause of greatly reduced contrast, which is difficult to avoid with a conventional instrument_ immediately becomes apparent when the fluorescence microscope is turned to imaging of intact and especially whole living tissues. This is blurring in the image from features that are out-of-focus, that is, situated away from the specimen plane in which the microscope objective lens is focused. Put another way, as the conventional microscope is focused away from a bright feature, the total amount of light detected or seen does not decrease, it is only spread out into a diffuse blur. The problem is particularly acute in fluorescence, compared to other microscope contrast methods, for two reasons. First, the light is emitted from, and also scattered by, the sample in all directions. Second, for the objective lens to collect the emitted light most efficiently it must have a large collection angle or numerical aperture (NA) and consequently a very short depth of focus (see Pawley [1995], Ch. 1).
B. Laser Scanning Microscopy A partial solution to the problem of contrast-degrading light scatter is found in the scanning optical microscope (SOM) [Sheppard and Wilson, 1984]. In its basic form, the SO M instrument consists of a scanned spot of illumination that is moved through the specimen and a detector that picks up light returning from the sample. In this way an image is built up point-by-point in picture-cells or pixels like a TV or PC screen image (usually the final display in a modern instrument). With this single-point illumination, scattered light from adjacent regions of the sample is greatly reduced. The detector collects light from the whole, or at least a large part, of the specimen for each image pixel. Both in-focus and out-of-focus light is collected and this ultimately limits the contrast, particularly in thick specimens. The use of laser light enables the scanning spot to be made as small as the theoretical resolution limits described above. Consequently the laser scanning microscope (LSM) has become the standard form. A characteristic feature of a nonconfocal fluorescence LSM is that the resolution is determined by the wavelength of just the excitation light. In a reversal of the case for conventional fluorescence, the fluorescence emission wavelength does not determine the resolution. Since the exciting light has a shorter wavelength than the fluorescence emission, the nonconfocal LSM has superior resolution to a conventional fluorescence microscope. Unfortunately, this improvement is often countered in thick specimens by the increased contrast-degrading scattering of the shorter wavelength light.
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C. Confocal Microscopy Even more important than the resolution improvement of the basic LSM (under ideal conditions) are the opportunities for implementing novel imaging modes through the ability to optically process each imaged point during the scan. Many new contrast techniques become possible by the use of split or differential detectors as well as further resolution enhancements using spatial filters l-Sheppard and Wilson, 1987]. One particular spatial filter, the confocal aperture (Figure 2), improves in-plane and axial resolution and also significantly reduces the contributions of light from out-of-focus regions in the specimen (see Pawley [1995]). The confocal aperture is placed in an image plane behind the objective lens. This, of course, means that the objective must focus the returning fluorescence to a fine spot and the optimum confocal effect requires this to be around half the resolution limit magnified by the lens power [Wilson, 1989]. The theoretical resolution of the confocal microscope is best understood as approximately the product of the resolving power of the optics used for both illumination and fluorescence detection. The term confocal simply means "co-focused" excitation and detection and typically the same objective is used for both in the convenient epi-fluorescence configuration, as used in most conventional microscopes. A fundamental difference is that the confocal performance is determined by the imaging characteristics of both excitation and detection. As a result, the confocal fluorescence LSM theoretical resolution is better by a factor of 1.4 or more (depending on the difference in excitation and emission wavelengths), times that for the conventional case [Brackenhoff et al. [1979]. To arrive at this figure it is necessary to consider the product of the illuminated focal-spot light density and a similar function representing an image of the confocal aperture projected into the specimen by the objective lens. The distance between the minimum of this function and the central peak is still the same as that described above for a single focused beam, but the peak is much sharper. The in-plane and axial separation distances in the equations above define the separation of two focused spots when the central peak of one overlies the first minimum of the other, that is, the half-width of the spot measured between the peak and minimum intensity. To indicate the increased resolution by sharpening of this feature in the confocal microscope, it is usual to report the value for the full width of this peak at half the maximum intensity (FWHM). Since the nonconfocal LSM relies on the excitation wavelength only, the confocal resolution improvement over the basic LSM is never better than 1.4 times and frequently a little less. This point might be noted for the later discussion of multiphoton excitation.
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FIGURE 2. A representation of a generalized confocal fluorescence laser scanning microscope. A single mode laser is used to illuminate a high NA objective lens, which forms a diffraction, limited focus within the specimen (appearing as a double-cone with a bright spot in the center). Fluorescence emitted from throughout the cones of focus is collected by the optics and passed towards the detector. A small confocal aperture, about the size of the focal spot magnified by the objective at its image plane, is placed before the detector and this blocks fluorescent light coming from above or below (dotted rays) the central plane of focus. Only in-focus light (solid rays) passes the aperture to contribute to the optical section image, formed by scanning the spot horizontally through the focal plane.
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For the biological microscopist, resolution comparisons are of secondary importance compared to the capability of the confocal system to exclude out-of-focus light, giving high contrast optical sections through even thick specimens. Serial sectioning at progressive focus positions routinely yields 3-D images or data sets. The confocal optical section thickness is not straightforward to calculate (or indeed accurately describe as it depends critically on the specimen being imaged) but it is similar in extent to the axial resolution FWHM.
4. Limitations of Confocal Microscopy Using two imaging components for excitation and detection provides the confocal optical sectioning effect and, under ideal circumstances, the 40% or so increase in spatial resolution, but this arrangement is also responsible for the most serious limitations of the system. It is clear that excitation and detection must be in precise registration within the specimen. Anything that causes even a slight nonoverlap will lead to a larger imaging "spot," lower resolution, and loss of image intensity. Chromatic (color) aberration is a particular problem of this kind. All optical components are designed to a given chromatic tolerance or correction. This means that they will usually focus light of different wavelengths to different locations in both the axial and in-plane directions. Traditional achromatic lenses can be fully corrected for two particular wavelengths but these are unlikely to correspond to even the peaks of the excitation or emission spectra required for most fluorescence applications. The highest quality Plan Apochromatic objective lenses will be fully corrected for three or possibly four wavelengths, again not necessarily exactly the colors needed for a particular dye or stain. It is thus usual to expect that the detection/excitation overlap or "confocality" is not perfect and, for a good objective, the deviation is typically around half of the resolution values described above [Fricker and White, 1992]. This almost always means that a compromise must be made in the setting of the confocal aperture, that is, by making it larger and thus reducing the confocal effect. Alternatively, the excitation is often prefocused with additional lenses before the objective so as to bring the two light paths in axial registration, little can usually be done about the in-plane problem except perhaps to reduce the field of view. Even the axial focus-correction is not ideal as it introduces additional inaccuracies such as spherical aberration because the objective lens is designed to receive a nonfocused (i.e., parallel) beam on the illumination side. Another problem has already been discussed, in part, above; that of fluorescence light from the specimen being scattered by the sample itself so it is not all focused through the confocal detection aperture. This leads,
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again, to loss of fluorescence signal in confocal microscopy, particularly when focusing deep into thick, noncleared or living specimens. Bending of light away from the correct focus of the objective lens results from refractive index variations in the specimen. This spreads out the excitation spot again decreasing the amount of light that can be effectively imaged back through the confocal aperture. The result is more loss of signal. In practice it is not usually possible to separate scattering and refractive effects in a biological specimen and both give rise to strong depth-dependent attenuation [White et al., 1996]. Lastly, the high-energy light required to excite short-blue or UV fluorescent probes is very damaging to living cells because of unintended absorption by intrinsic components. When excited, molecules can combine with oxygen to produce highly reactive free radicals that will quickly degrade other dye molecules (photo-bleaching) or a living specimen itself (photo-toxicity) and/or physical photo-destruction of the sample. Several important UV-excited fluorescent probes for DNA, calcium, and other ions together with a range of useful auto-fluorescent natural compounds do not yet have fully equivalent substitutes that work with less-energetic longer wavelengths. To summarize, there are three key areas where the 3-D LSM might usefully be improved: 1. Removing the dependence on confocal alignment for optical sectioning performance. 2. Improving the detection efficiency for fluorescence light coming from highly scattering or refractive specimens, particularly in deep samples. 3. Using lower-energy long wavelength light, thus reducing photodamaging effects, while imaging common and well-characterised fluorescent probes. Despite these problems confocal microscopy has furnished cell biologists with a vast range of microscopy methods and protocols for imaging intact and live specimens [Pawley, 1995; Paddock 1999]. We can now turn to the further advances and benefits that multiphoton microscopy has added to this field. D. Multiphoton L S M
Multiphoton microscopy provides improved high-resolution optical sectioning images of widely used fluorescent probes by providing solutions to the three areas for improvement of the confocal LSM [Denk et al., 1990]: 1. Optical sections are obtained without the use of a confocal aperture or any spatial filters.
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FIGURE 3. Jablonski energy diagram of two-photon excitation. A fluorescent molecule is excited to a high energy excited singlet state by the absorption of two photons of excitation light that interact together within a time period of < 10-16 s (i.e., essentially simultaneously). The combined energy of two low-energy (longer wavelength) photons is sufficient to excite a fluorescent molecule that might normally be excited by the energy of a single photon of approximately half the wavelength. The process of fluorescence emission is the same as for single-photon excitation (Figure 1), except that the emitted light appears as a longer wavelength than the illumination light.
2. The detection optics can be configured not to produce an image of the excitation spot but merely to efficiently collect light returning from the sample through a wider range of angles. 3. Lower-energy, farred, or infrared wavelengths are used to excite common fluorescent probes using the combined energy of two or more simultaneously absorbed photons.
5. Multiphoton Excitation All the essential features and benefits of this new L S M technique are derived from the novel way that the fluorescent dye molecules are excited (Figures 3 and 4). In contrast to normal or single-photon excitation, multiphoton absorption requires that two or more photons interact with the fluorescent molecule within an extremely short period of time ( < 10-16 S). This is such a short time that there is no physical meaning to the excited state of the molecule between the arrival of the first and second photons; it is thus considered as a virtual state. The photons can be considered as interacting simultaneously with the combined energy equivalent to a single photon of
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FIGURE 4. A physical demonstration of one- and two-photon excitation. A cuvette of fluorescent solution shows the difference between single- and multiphoton excitation. In the single-photon case (upper), fluorescence is excited throughout the focus cones through the specimen. A confocal aperture is needed to remove the signal from either side of the central focal plane. In the multiphoton case (lower) only the very center of the focus has sufficient laser light power (at twice the wavelength of the single-photon example) to induce the solution to fluorescence (only a small bright spot of fluorescence is seen). No fluorescence is produced on either side of the focal point. B. Amos, LMB, Cambridge, UK.
half the wavelength. It is thus quite normal with this method to excite fluorescent probes with light of approximately twice the usual wavelength and produce fluorescence with a shorter wavelength than the illumination. This is, of course, not a new kind of fluorescent emission, since the important criteria is that the fluorescence is of lower energy than the excitation, and is identical to the emission process for the single-photon case. The emitted light has, indeed, lower energy than the combined energy of the two or more absorbed photons.
6. Fluorescence Optical Sectionin9 by Nonlinear Excitation It is not immediately clear how multiphoton excitation gives rise to optical sections without the use of a confocal aperture to block out-of-focus light until the statistical nature and geometry of the excitation process are examined.
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For single photon absorption, the likelihood or probability of a given molecule being excited by the incoming light is directly proportional to the power or light intensity. The light intensity can be seen as the brightness of the beam when it passes through the focus of the objective lens rather like two cones joined at the point (Figure 2 and Figure 4). There is no more light at the center (point) of the focus than in the spread-out beams above and below, it is just concentrated into a smaller volume. For this reason the total amount of fluorescence excited in any plane that is at right angles to the beam will be a constant. There is no preferential excitation at the in-focus (central) plane in single-photon excitation. As stated before, only the light from the central plane is allowed to contribute to the image by the confocal aperture which blocks or "throws away" the out-of-focus light. For multiphoton absorption, the probability of a molecule encountering a single photon is again proportional to the power of the laser light exactly as before. This is true for encounters with any of the photons in the illumination. However, if we wish to consider the likelihood that a fluorescent molecule will encounter two photons (i.e., one photon and then another almost simultaneously) we must take the product of the single-photon absorption probability [Denk, 1990; Stelzer et al., 1994, see also Pawley, 1995, Ch. 28]. In this way the fluorescence intensity produced (for the nonconfocal geometry) is directly proportional to the squared or cubed power of the illumination intensity for two- or three-photon excitation respectively. Although the total amount of light in all planes through the cones of focus is still constant, the excitation efficiency is very much greater in the center (i.e., in the focal plane) because of the nonlinear (second or third order) effect. This generates a thin plane of fluorescence, an optical section, entirely from the excitation light (Figure 5). In nonconfocal multiphoton microscopy the fluorescence detection does not determine the optical sectioning performance. The section thickness is approximately equal to the F W H M axial resolution. The wavelength of the original farred or infrared illuminating light must be used in this calculation, rather than the equivalent "wavelength" of the combined photon energies. For this reason, multiphoton microscopy has a lower resolution than single-photon confocal microscopy for a given fluorescent probe. In defense of the multiphoton performance we can say that the 40% "super-resolution" in confocal images compared to conventional microscopy is rarely achieved in thick biological specimens, whereas an equivalent sharpening with twophoton excitation is often observed. 7. Laser Illumination Requirements
Although the multiphoton excitation probability reaches a maximum within the optical section it must, of course, be high enough to produce a detectable
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FIGURE 5. Representation of a generalized multiphoton laser scanning microscope. A single mode, near-IR pulsed laser is focused by an objective lens into a double cone of illumination through the specimen. The long wavelength, high power and pulsed character of the laser is such that fluorescence is generated only at the center of the focused beam in the focal plane. Fluorescence light emitted from the resultant optical section and collected by the objective lens is passed efficiently to the detector (solid rays). Fluorescent light scattered by the specimen (an example fluorescent ray is shown scattered into several dotted rays) and passed through the objective is also detected and contributes to the image.
signal with a useful signal-to-noise ratio. This implies t h a t a sufficient laser intensity m u s t be used w i t h o u t excessive h a r m to the specimen. This is achieved by a c o m b i n a t i o n of three key steps: 1. Use of a high p o w e r laser with a g o o d gaussian b e a m profile t h a t can be t u n e d to the m o s t efficient w a v e l e n g t h for m u l t i p h o t o n excitation of the c h o s e n probe.
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2. Producin~ the laser outm~t in .~hart m~l~e,~ ~a a,~ to maximiTe the nonlinear excitation probability (the so-called short-pulse advantage). 3. Diffraction-limited focusing of the laser to the smallest spot possible by an objective lens with high transmission and the minimum of distortion to the spot or pulse characteristics of the beam. The laser power typically used for two-photon absorption is at least 100 times that required for single-photon excitation and around a further 10 times more for three-photon work. Even this amount of power would be insufficient from a constant power (CW) laser for practical multiphoton excitation. Increasing the average laser power further would quickly produce harmful effects at the specimen so the light energy is compressed into extremely short pulses with a repeat-rate that results in no increase of the time-averaged power. The nonlinear absorption (by the square or cube of the laser power) is sufficient to produce multiphoton excitation at these power levels if the pulses are around 100 fs (or less) in length repeated at 80-100 Mhz. This corresponds to the shortest pulse that can be transmitted reliably through a microscope without excessive distortion (see Brackenhoff et al. [1995] for measurements of pulse length and laser power at the specimen). The repeat rate corresponds to around 10 ns, long enough for most fluorescence molecules to go through an excitation-relaxation cycle (typically a few ns). A little extra time also ensures that after each pulse the fluorescent molecules are all returned to the ground state, thus avoiding one cause of dye saturation. Pulses of several ps can, in principal, also be used but require a disproportionately greater laser power (about 3-30 times) to produce the same fluorescence signal as 100fs pulses. With extreme levels of power (typically several hundred times more) two-photon excitation with a CW laser can be demonstrated. In general, the multiphoton excited fluorescence intensity I for a particular probe is given by p, I o c ~
(wR) " - 1
where P is the average laser power, n is the number of photons absorbed for each fluorescence photon, w is the length of the laser pulses, and R is the rate at which the pulses are repeated. The "short pulse advantage" increases from two- to three-photon excitation due to the inverse power dependence o n w.
Efficient focusing of the laser requires more or less the same characteristics of a good objective for single and multiphoton cases. However, for the nonlinear modes the laser pulses can be significantly stretched out in time
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by a lens with high dispersion. This is due to group velocity dispersion (GVD), a process analogous to chromatic aberration operating on the distribution of wavelengths in each light pulse [Guild et al., 1997]. Since most common objectives are designed for visible nonpulsed illumination, dispersion can be a major loss of power through pulse-broadening. Even so-called "IR" lenses are often only corrected at about 780nm for IR Nomarski imaging, having poor performance at longer wavelengths including severe pulse-broadening dispersion. Lens and microscope manufacturers are starting to respond to the requirements for optics that are better corrected for the special demands of multiphoton excitation with pulsed lasers. For the most demanding applications precompensation of the laser pulse (prechirping) may be implemented. 8. Practical Laser Sources for Multiphoton Excitation Before the advent of solid-state pump lasers, large argon-ion lasers were used, originally to pump pulsed dye-lasers. For commercial multiphoton microscopes, the laser of choice (at the time of writing) is the Titanium Sapphire (TiS) pulsed system (Spectra Physics Inc. and Coherent Inc., USA). The system consists of three lasers. One or more infrared laser diode(s) provide the initial light source, which is frequency doubled in a solid state pump laser. The pump uses NdYVO 4 as a gain medium and a nonlinear crystal (LBO) to produce a green output (532 nm) of 5 to 15 W. Pump laser energy is used to excite a titanium-doped sapphire crystal (TiS) within the cavity of the pulsed-laser unit. The crystal fluoresces when excited by the green light and the cavity is tuned to provide high gain laser output at the required wavelength. In this way the tuneable TiS laser can provide up to 2 W peak output and wavelengths from around 680nm to about 1080nm. These permit efficient excitation of fluorescent probes normally excited between about 320-560nm (by two-photons) or (less efficiently) around 250-350nm by three-photons. Fixed-wavelength versions of the TiS laser, with an integrated pulse laser, are also available and for certain well-defined applications they provide a simpler (though not significantly less expensive) solution. A number of manual adjustments are required to correctly operate the tuneable systems, but rapid advances in system-integration, semi-automatic and PC-controlled tuning and/or pulse control are being made by the principle manufacturers. Alternatives to the TiS laser have been used successfully for biological microscopy. A diode pumped NdYlf laser (Coherent (USA)/Microlase, Scotland) provides a useful output (500mW) at 1047 nm at 160fs pulses, which permits more efficient three-photon excitation equivalent to about
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350nm_ llsin~ this laser, three-nhoton excitation of DAPI (a DNA stain) and two-photon excitation of two other probes has been demonstrated in living specimens [-Wokosin et al., 1996a, 1996b]. Although the system includes an optical fiber (allowing some remote siting of the main unit), postfiber pulse compression is required and this results in a system with several critical alignments. Several new lasers are set to appear over the coming year or so. Some biological work (imaging a calcium probe--Calcium Green) has already been demonstrated with a CrLiS laser operating at 850 nm [Svoboda et al., 1996]. 9. Nonimaging Detection Optics
Because, unlike the confocal case, the light-collecting optical components need not make an image of (i.e., focus) the fluorescent spot onto a small point-aperture, the characteristics of the detector system can be more efficiently concentrated on gathering as much of the fluorescence signal as possible. The simplest way of maximizing this property is to site the photodetector(s), usually photomultipliers (PMTs), as close to the specimen as possible. It is not usually practical to mount them immediately adjacent to the sample as some wavelength-selecting filters must be interposed to block illumination light or unwanted fluorescence wavelengths from contributing to the detected signal. This means that in most practical instruments a high NA conventional objective lens is used both for focusing the nearinfrared laser and collecting the visible fluorescence. Due to its high degree of corrections in the visible region it is not theoretically maximally suited for either task, although some corrections are important to achieve a small excitation spot and high resolution. The detectors are usually placed between the scanning system and objective [Piston et al., 1992, 1994]. Fluorescence light does not pass back through the scanning system but is efficiently captured by these "scattered light" detectors as they view the entire specimen, rather than being focused onto just the excitation spot as in confocal microscopy. It is certainly the case that much of the high-angle scattered fluorescence, particularly from deep in thick specimens, does not pass back through the objective. However, the combination of high NA and closer proximity of the PMT(s) to the exit pupil of the objective allows a large aperture detector to gather fluorescent light more efficiently than the confocal geometry. This is true for light coming directly from the focal plane but is an important way of detecting fluorescent emissions after they have been scattered by the specimen. Many important problems in biological microscopy require the imaging of cells and their internal structure and function within their normal
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FIGURE 6. Microscopy of cellular and matrix components in cartilage. Low-resolution optical sections of human intervertebral disc stained with a general histological stain for connective tissues: Thick collagen fibers (a and b) pass through extensive areas of extracellular matrix (ECM) proteins (c). High resolution optical sections of living rat intervertebral disc cartilage: d: Autofluorescent ECM fibres approximately 20#m below the surface of the cartilage, e and f: Living chondrocytes (cartilage cells) positively stained with CMFDA appear between the autofluorescent fibers at approximately 40 #m and 60 #m into the cartilage. Plan Apo 60 x 1.2 NA water immersion objective. 770 nm TiS excitation, > 400 nm emission, using external scattered-light detectors.
environment inside a living, intact tissue. Often, the noncellular parts of the tissue consist of matrix proteins that strongly scatter both the excitation and fluorescence emission light. Multiphoton microscopy with scattered-light detectors is particularly effective at providing the highest contrast images of cells in vivo particularly in thick biological tissues (e.g., Figure 6). For samples that can be imaged throughout their entire depth, a detector may be mounted after the specimen in the "transmission" configuration. With high quality infrared-blocking filters it is possible to collect
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fluorescence emitted away from the objective lens (and so not detected by the usual epi-configuration). This is the correct description of so-called "external" detectors, although the term is often used to describe any large-area detector of nondescanned light (i.e., light that does not pass back through the scanning system). Both epi- and transdetected signals can be combined to increase the signal-to-noise of the resulting image. Many other detection arrangements are possible and it is likely that developments in this area will significantly improve the performance of multiphoton microscopes for even better images from within thick nontransparent samples. 10. Laser Interactions with the Specimen The amount of laser light that penetrates a thick biological specimen to form the focused spot is critically dependent on the optical character of the sample through which the illumination has to pass to reach the area of interest. Absorption, refraction (light bending) and scattering of laser light by the specimen seriously limits the amount of power available to image regions more than a few tens of micrometers into most preparations. Biological material does not usually absorb a significant amount of the illumination. However, fluorescent molecules in the bulk of the specimen before the optical section can absorb some of the light in single-photon microscopy. In multiphoton excitation, only the optical section experiences the kind of light energy required for significant absorption by probe molecules. Regions above (and below) the focal plane only encounter the long wavelength, low-energy light and do not contribute to this particular depth-dependent attenuation. Scattering and refraction of light, on the other hand, do occur throughout the sample and for thick nontransparent preparations this is a major loss of signal at depth. There are many different types of scattering from different structures of different sizes. For small particles up to the wavelength of light in size, the amount of light redirected by scattering away from the focus is inversely proportional to the fourth power of the wavelength. This attenuation effect is thus greatly reduced for near-infrared light compared to the shorter visible wavelengths used for single-photon excitation (see Jacobsen et al. [1994]). Microscopy applications in plant science research have a particularly strong requirement for scattered light detection. It is increasingly important (for example) to study the 3-D morphology of intact developing structures in normal and model organisms genetically modified by mutations (Figures 7 and 8).
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FIGURE 7. Multiphoton microscopy in developmental studies of plant seeds. A young embryo and peripheral endosperm (a) and the chalazal endosperm (b) in a developing Arabidopsis seed. M. Spielman, Bio-Rad BMU.
Apart from the contrast or signal-degrading aberrations described above the most important interactions of the high-power laser light with a fluorescently stained specimen are the three kinds of absorption events that may occur: 1. Multiphoton absorption by the fluorescent probe itself. 2. Multiphoton absorption of naturally occurring molecules. 3. Single-photon absorption by the specimen and possibly (rarely) by the fluorescent probe. It is important to understand that all of these processes can, in principle, lead to localized heating, but at the laser scanning rates usually employed (a /zs or so per pixel) the small amount of heat generated in biological preparations is adequately dissipated (see Pawley [1995], Ch. 28). Excited molecular species (by whatever absorption event occurred) can produce free radicals that damage the dye and sample by combinations of photobleaching, photo-toxicity, or other physical photo-damage. The processes of fluorescent probe photo-bleaching with multiphoton excitation are not completely understood, though they certainly involve the same processes associated with the single-photon case, perhaps with additional interactions
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FIGURE 8. Multiphoton microscopy of developing pollen in plants. Pollen development sequence in plant anthers. Wild type: A small flower bud with two anther primordia (a). Optical section through one of the four locules comprising an anther showing the central pollen mother cells before meiosis (bright spots are nuclei) (b). Optical section through two locules showing central mother cells at the tetrad stage of meiosis encased in their callose wall (c). Maturing pollen grains where the meiotic products become released from their callose wall and further develop in the anther locule (d). Experimental mutants: Young anthers of the esp mutant before meiosis (e), anther wall layers are discontinuous and the two internal layers are often missing. Later stage in the mutant development (f) showing signs of cell degeneration, which eventually results in incomplete meiosis and no pollen. C. Canales, Bio-Rad BMU.
with the excited state. T w o - p h o t o n induced p h o t o - b l e a c h i n g is a process t h a t in some circumstances can increase by the third or fourth p o w e r of the laser intensity a n d m u c h research is currently u n d e r w a y to explore the particular processes responsible. W h a t e v e r the m e c h a n i s m , it is clear that, at routine laser powers used for imaging, the i m p o r t a n t p h o t o - d e s t r u c t i v e effects have their m a j o r contribution from the first two processes listed a b o v e (e.g., F i g u r e 9), are thus limited to the optical section or even just the very center of this region. This does not necessarily m e a n that on a section-for-section basis bleaching is always less for m u l t i p h o t o n excitation than the single p h o t o n case, but the latter exhibits any d a m a g i n g effects t h r o u g h o u t the specimen for each plane
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FIGURE 9. Comparison of photo-disruption in living unstained cells for pulsed and nonpulsed laser illumination. Cultured HeLa cells were imaged in a laser scanning microscope using the non-optical-sectioning transmission contrast mode. The specimen was continually scanned at one frame per second and images collected at timed intervals. Data were collected at various laser power levels and with approximately 100fs pulses or continuous power illumination. Vesicular structures derived from the cell membrane and outgrowths or "blebs" of the plasma membrane itself develop as a time-dependent indicator of laser-damage. The cells were unstained. No significant damage (vesicular appearance) was observed over time (indicated in seconds) with continuous (CW) illumination at 200mW (upper series). Increasingly rapid appearance of vesicles was observed with laser power delivered as 100 fs pulses (200 mW time series is shown in the lower sequence). The laser power dependence of damage indicates that the process is a second- or higher-order process consistent with multiphoton absorption.
scanned. Out-of-plane d a m a g e in the confocal microscope m a y also lead to artifacts within the optical section. F o r these reasons, m u l t i p h o t o n excitation usually gives less d a m a g e overall within the volume of the specimen w h e n 3-D serial-section images and 4-D (time-lapsed 3-D images) are acquired. T h r e e - d i m e n s i o n a l imaging of D N A dyes such as D A P I and H o e s c h t (Figure 10) by m u l t i p h o t o n excitation avoids the excessive UVd a m a g e to samples that occurs with single-photon a b s o r p t i o n outside of the optical section. W h e n the dye being imaged is free to m o v e or diffuse in and out of the optical section (such as a physiological probe for intracellular calcium or other ion) m u l t i p h o t o n excitation and efficient (scattered light) detection can give significantly reduced p h o t o - b l e a c h i n g (Figure 11).
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FIGURE 10. 3-D microscopy of chromosomes stained with the "UV" DNA dye Hoescht by multiphoton excitation. Two different stages of chromosome condensation during cell mitosis (A and B show 3-D projections of serial optical sections). Both excitation of the fluorescent probe and photo-bleaching are restricted to the optical plane in two-photon microscopy. This allows 3-D imaging by serial optical sectioning without photo-bleaching the fluorescent dye molecules when they are not contributing to the acquired image.
Three-photon absorption by D N A (e.g., Wokosin et al. [1996], N A D ( P ) H (e.g., Piston et al. [1994b]) and/or proteins may also occur if particularly high laser powers are used, or possibly by two photon excitation using the shortest possible wavelengths of pulsed laser. This will almost always have some negative effect on living preparations. Additionally, these compounds may contribute to an elevated background of auto-fluorescence but this can usually be efficiently filtered out of the detected signal. If the laser power and frequency of scanning are set to avoid these problems, multiphoton imaging of functional probes in living cells often within their natural intact tissue is routinely possible (Figures 12 and 13). Single-photon absorption is not a significant problem at the laser power levels used for most biological specimens, including many living preparations, provided there are not any major absorbing species and the highest
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FIGURE 11. Laser scanning microscopy of the intracellular calcium probe Indo-1. Singlephoton confocal excitation of Indo-1 using 351 nm UV laser illumination typically gives rise to photo-bleaching of the fluorescent probe (a) both within and outside the optical section. The freely diffusing dye exhibits a strong time-dependent loss of fluorescence, decreasing the signal-to-noise ratio in the acquired data. With multiphoton excitation at 750nm and approximately 100fs laser pulses, Indo-1 can be followed for time-lapse studies with substantially less photo-bleaching (b). Both graphs represent the calcium sensitive signal (emission around 480nm) from Indo-1 extracted from a region of the optical section and plotted over time. efficiency pulses (100 fs or less) are used to a v o i d excessively high average laser power. C h l o r o p h y l l a n d related p h o t o - p i g m e n t s (in green plants, algae a n d so on), are g o o d e x a m p l e s of e n d o g e n o u s c o m p o u n d s that can exhibit b o t h m u l t i p h o t o n a n d s i n g l e - p h o t o n e x c i t a t i o n in the farred w a v e l e n g t h r a n g e b e t w e e n 700 a n d 8 0 0 n m . It is advisable in such p r e p a r a t i o n s ,
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FIGURE 12. Multiphoton microscopy of glutathione in living Arabidopsis roots. The complex and often dense morphology of the young Arabidopsis plant root is clearly imaged throughout the tissue by multiphoton excitation (c). Multiphoton optical section through epidermal cells of a live Arabidopsis root (a) 5 min after labeling with monochlorobimane (MCB) excited at 770nm. The bimane dye is initially nonfluorescent until it binds to glutathione (GSH) in the cell cytoplasm yielding a fluorescent GS-B conjugate. GSH is a tripeptide that is involved in detoxification of xenobiotics and possibly other environmental stress responses. In this early time point, the cytoplasmic staining appears as a thin layer around the large vacuole of each cell. (b) longitudinal section through the midplane of a live Arabidopsis root, near the tip, 30 min after labeling with MCB. The fluorescent GS-B conjugate has been transported into the vacuole of each cell. A. J. Meyer, M. D. Fricker, and N. S. White, Bio-Rad BMU.
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FIGURE 13. Multiphoton microscopy of probes for subcellular compartments and functions in living cells. Transgenic probes based around expression of the green fluorescent protein (GFP) can be efficiently imaged by multiphoton excitation. ER-targeted G F P in Arabidopsis plant leaf epithelium can be followed over many minutes or hours with excitation at 800 nm (a-d). Cultured HeLa cells transfected with the calcium indicator Cameleon probe (e). This is a hybrid of Cyan-FP and Yellow-FP with a calcium sensitive linker of calmodulin and M13 that emits blue and yellow fluorescence when excited at 790 nm. This choice of wavelength gives the most efficient excitation of CFP with minimal direct excitation of YFP. The ratio of these two emissions depends on the amount of blue emission from the CFP that is able to excite the excitation peak of YFP by fluorescence resonance energy transfer (FRET). Calcium binding to the linker region causes the molecule to flex, changing the separation of the CFP and YFP and the efficiency of FRET. The expressed Cameleon has no specific targeting sequence and ends up in the cytoplasm, and is believed to be too big to enter the nucleus. Specific targeted versions of this, and related probes, allow the determination of calcium levels in specific cellular compartments.
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especially where a small amount of added fluorescent probe or a weakly excited dye must be used, to choose a system working at wavelengths above 900nm [Wokosin, 1992, 1994a]. However, if the highest powers available from current lasers are used with thin preparations, some single photondamage may become a problem if the dissipation of heat by diffusion is insufficient.
11. Multiphoton Instrumentation and Protocols: Evolution or Revolution? Historically LSM developments have progressed largely in the order described above. It is, perhaps, unfortunate that this has resulted in commercially available multiphoton instrumentation and protocols being largely seen (and indeed often presented) as an evolution of confocal microscopy. Although the two techniques can be used in combination (discussed below) the particular advantages gained are generally more than outweighed by the loss of the improved detection benefits of a completely nonconfocal arrangement. For most applications it is more useful to re-think some of the central tenets of practical confocal microscopy and replace them with more appropriate guiding principles for multiphoton excitation. Again some key areas can be identified: 1. Choice of useful fluorescent probes (particularly in combination) for particular applications. 2. 3. 4. 5. 6.
Choice of appropriate optical components. Optimization of the excitation optics. Fluorescent selection and excitation barrier filters. Optimized detection (discussed at length above). Management and adjustment of optical section thickness.
12. Choosin9 Fluorescent Probes for Multiphoton Microscopy The first issue to resolve here is what laser wavelength is available or chosen. Due to the high cost, it is not yet realistic to have more than one pulsed-laser available on a particular LSM system, therefore, a single wavelength must be used to excite the probes of interest in a given sample. A compromise situation could be imagined where confocal excitation of one probe (with a visible laser) was used simultaneously or sequentially with the multiphoton laser but this can become complicated by the objective lens illuminating different planes due to chromatic aberration. In any case it is highly desirable to make use of the benefits of multiphoton imaging for all the probes used in a particular study. Tuneable lasers allow significant flexibility in excitation but they require several seconds to tune reliably between wavelength settings.
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Multiphoton excitation spectra tend to be flatter and broader than equivalent single-photon curves and thus it is not always necessary to be exactly at the peak of absorption; sometimes a longer wavelength can be chosen to help minimize damage to the specimen and maximize penetration. This has disadvantages as well, since it is rarely possible to avoid exciting unwanted molecules with peak absorption in nearby parts of the spectrum. Unlike most multichannel confocal microscopy, blocking of unwanted auto-fluorescence and discrimination between multiple probes must be accomplished entirely by filters in the emission path. All the probes to be imaged will be excited all of the time and thus sequential excitation methods cannot be used to avoid spectral overlap. Some combinations of dyes that work extremely well with conventional sequential imaging are more challenging to separate reliably with multiphoton excitation. A typical example is the use of DAPI for DNA staining combined with fluorescein or similar green probes. When simultaneously excited the long green/yellow tail of the DAPI signal "bleeds into" the green fluorescence channel. Three things can be done to minimize this effect: First, it is always best to choose probes that have minimal or no overlap of emission spectra. For this reason DAPI is best used with yellow- or red-emitting probes. Second, the best compromise of excitation wavelength is necessary to minimize the ratio of bleed-through to signal in the long emission channel while maintaining an adequate signal in the shorter detection channel. Third, a general rule of thumb should always be adopted for multiple staining (equally relevant to all other fluorescence configurations as well); the staining levels of each probe should be increased very roughly in proportion to the wavelength of emission. In this way the longer detection channels will be operated with decreasing PMT voltage (gain) and thus will amplify progressively less the bleed through from shorter emissions in the preceding channels. 13. Choice of Optical Components A new criterion must be met for all components that are to be inserted into the multiphoton laser illumination path, that of minimal distortion of the pulse shape. It should also be noted that, because of the complete dependence on the excitation path, single components that introduce aberrations may result in a disproportionately larger effect than when inserted into just one of the two imaging paths of the confocal microscope. An example of a troublesome component is the acousto-optic modulator (AOM). While being reasonably efficient at blanking or attenuating CW beams, these devices introduce serious distortions to both the wave-front and pulse shape, requiring extensive corrections or control. A more satisfactory alternative is found in the Pockels Cell (see Pawley [-1995], Ch. 28) which has much less
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effect on the beam as it changes the polarization state (and is thus used with an appropriate analyser for blanking or attenuation) rather than the phase properties. The multiphoton pulsed beam presents another opportunity to implement an attenuation function that is potentially "cleaner" in its interaction with the primary beam properties; that of pulse selection or "picking" to transmit a portion of the laser power by blocking some of the pulses. This has the advantage of providing a simple attenuation of fluorescence excitation that is linear with the output signal (rather than with the square or cube for the equivalent blocking of time-averaged power by neutral density filters or an AOM device). 14. Optimizin9 the Illumination Path Multiphoton excitation allows the illumination and detection paths, at least in theory, to be completely separated. Even in the widely used epi-configuration most of the excitation path (except the shared objective lens) need not used by the fluorescence emission if scattered light detectors are optimally implemented. Under these circumstances, optics specially designed for farred or infrared pulsed light could be used for a dedicated system. By optimizing the design, particularly with reflecting-only components (to avoid excessive dispersion) and by using efficient surface coatings (e.g., those developed for the internal laser components), a multiphoton illuminator can efficiently transmit the energy of substantially lower-power lasers and still achieve acceptable signal levels. This would provide a strong incentive to system manufacturers to press for lower-cost pulsed lasers, paving the way for affordable multiple-wavelength solutions. Overcoming the limitations of losing the capability for confocal microscopy, for example, by providing alternatives for common extra-long wavelength single-photon dyes or section thickness control by changing the aperture size, would help to realize a practical, dedicated, multiphoton instrument. A more expensive compromise is to use independent, optimized illumination paths for confocal and multiphoton illumination, switched by an accurate interposed beam-splitter or reflector. Again, the multiphoton method provides an advantage for this solution as moveable components need not maintain the highest accuracy in the detection path when no alignment with a confocal aperture is required. 15. Adjustment of Optical Section Thickness and Resolution Due to the absence of a variable confocal aperture, the multiphoton system must make use of an alternative method of changing or tuning the optical section thickness to the requirements of the application. The simplest is to choose an objective with a suitable NA. This has less of an effect on the level
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of fluorescence excitation than in single-photon microscopy since the reduction in peak intensity of the focused spot is balanced by the increase in the size of the spot. The lowered fluorescence detection by the reduced collection angle is not so easy to compensate. The full collection efficiency of a high NA objective can be used with an increased spot size for thicker sections by reducing the diameter of the illumination beam so as to underfill the objective lens. It is important to fill the objective lens entrance pupil normally to obtain the full NA when the maximum resolution (i.e., smallest spot) is required. To fulfill both of these requirements an efficient, variable beam expander is required that maintains the desired optical section and/or resolution characteristics with a range of objectives without losing laser power by excessive overfilling. This component is also useful to optimize beam delivery for single-photon excitation, but for multiphoton systems this should ideally be of a reflecting design, to avoid excessive pulse-spreading by dispersion in refractive glass optics, with an optimized reflection coating for the wavelengths delivered by the laser. II. FUTURE PROSPECTS
There is every reason to expect that the rapid advances in multiphoton LSM techniques will continue to provide more appropriate imaging tools for biologists to work with intact, and therefore relevant, preparations. The simplified optical arrangement of multiphoton microscopes will make these instruments more widely available at affordable cost only when new inexpensive solid state lasers are produced in sufficient volume. Multiline lasers will fill the gap in the current range of applications by increasing the separation of multiple probes while allowing quantitative imaging of excitation-ratio ion probes. Many new nonlinear imaging modes (e.g., 2nd harmonic microscopy) will become routine when multiphoton interactions with unstained biological specimens are better understood. In parallel with these developments, it is likely that the already growing interest in spectral analysis of intrinsic autofluorescent components will enable complex structural, metabolic, and physiological processes to be followed in intact, unstained living material. REFERENCES Brackenhoff, G. J., Blom, P. and Barends, P. (1979). Confocal scanning light microscopy with high aperture immersion lenses. J. Microsc. 117, 219-232. Brackenhoff, G. J., Muller, M. and Squier, J. (1995). Measurement of femto-second pulses in the focal point of a high-numerical aperture lens by two-photon absorption. J. Microsc. 179, 253-260.
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Index
A Acousto-optic modulator (AOM), 274-75 Algebraic topology, 46 Analogies between theories, 5 Annular dark-field (ADF) imaging: applications, 198 description of, 149-50 Annular dark-field STEM (ADF-STEM): development of, 152 future prospects, 198-99 image processing, quantitative, 190-97 longitudinal coherence, 170-81 resolution and information limit, 181-90 transverse incoherence, 153-70 Anomalous edge elements, l l0
B Bayesian methods, 195-97 Bloch type stray field, 207, 212 Bloch waves: calculations with TDS, 178-81 dynamical scattering using, 165-70 Boundary conditions and sources, 37-38 Boundary operator, 54-55
C Cell-complexes, 46-49 Chains, 52- 54 boundaries, 54-55
cochains, 57-60 Charge content, 25 Charge-current potentials, 30 Charge flow, 25 Chromatic aberrations, 185-89 Classification diagrams, defined, 6 Classification diagrams, Tonti, 6-7 for electostatics, 38, 40 of global electromagnetic quantities, 25-27, 59, 69 for heat transfer equation, 39-44 of local electromagnetic quantities, 24 for magnetostatistics, 38, 41 of physical quantities, 14 of space-time, 16 Classification schemes, 6 CLEAN algorithm, 194-95 Cochains, 57-60 -based field function approximation, 105 Coboundary operator: defined, 64-65 properties of, 65-67 Coherent imaging: See also Longitudinal coherence difference between incoherent imaging and, 150-52 Collector aperture, 162 Collocation method, 134 Complex amplitude, 150 Confocal microscopy, 253-56 limitations of, 255- 56 Constitutive equations, 19-20, 33-37 discretization errors and, 35-37 279
280
INDEX
Constitutive relations: discrete representation of, 69-72 discretization, strategies for, 97-105 error-based discretization, 103-5 field function reconstruction and projection, 99-103 global application of local, 98-99 Hodge star operator, 97-98 Continuous representations, 72-86 differential forms, 74-75 differential operators, 78-81 spread cells, 81-85 weak form of topological laws, 85-86 weighted integrals, 75-78 Contrast transfer function (CTF), 158, 181 Convergent beam electron diffraction (CBED), 156, 160, 177 Coordinate maps, 75 CTEM, 153-54, 160 Current, external orientation and, 23
for continuous representations, 72-86 for fields, 55-63 for geometry, 45-55 for topological laws, 63-69 Discrete Surface Integral (DSI) method, 122-26 Discretization errors, constitutive equations and, 35-37 Discretization step, 2 Discretization strategy, reference: constitutive relations discretization, strategies for, 97-105 edge elements and field reconstruction, 105-12 space-time domain discretization, 87-89 topological time-stepping, 89-96 Dynamical scattering: longitudinal coherence and, 173-74 using Bloch waves, 165-70
D
Edge elements, field reconstruction and, 105-12 Elastic scattering, 162-64 Electric charge, law of conservation of, 28, 29-30 Electromagnetic potentials, 30 Electromagnetism, equations of, 18-19 Electron trajectories, numerical calculation of, 217-20 Element mesh, 111 Elements, 110, 131 Error-based discretization, 103-5 Exterior differential, 79
Deconvolution methods, 192-97 Bayesian, 195-97 CLEAN algorithm, 194-95 multiplicative, 192-94 subtractive, 194 Delaunay-Voronoi meshes, 98 De Rham functor, 102 Differential forms, 74-75 Differential operators, 78-81 Discrete Green's formula, 83 Discrete representations: for constitutive relations, 69-72
E
INDEX
External orientation. See Orientation, external and internal
F Factorization diagrams, 6, 34, 35, 71-72 Fan-shaped stray field, 207, 210-12 Faraday's law: DSI discretization of, 123-24 induction, 18, 22, 28, 29 orientation and, 9, 10 topological time-stepping, 89, 90-91 Ferromagnetic domain boundaries, imaging of. See Imaging of ferromagnetic domain boundaries using PEEM Field function reconstruction and projection, 99-103 Field reconstruction, edge elements and, 105-12 Fields, 55- 56 cochains, 57-60 defined, 1 limit systems, 60-63 physical field problem, 1-2 Finite cell complex K, 47-48 Finite difference (FD) method, 112 categorization of, 2 Finite Difference-Time Domain (FDTD) method, 98, 11318, 121-22 Support Operator Method (SOM), 118-21 Finite Difference-Time Domain (FDTD): method, 98, 11318, 121-22 Finite element (FE) method: categorization of, 2
281
defined, 131- 33 time-domain, 134-36 time-domain edge element, 136-39 time-domain error-based, 139-40 Finite Integration Theory (FIT) method, 126- 31 Finite volume (FV) method: categorization of, 2 Discrete Surface Integral (DSI) method, 122-26 Finite Integration Theory (FIT) method, 126-31 Fluorescence microscopy, 249-52 conventional, 251 optical performance, 251 out-of-focus blur, 252
G Galerkin method, 134 Gauss's divergence theorem, 20 Gauss's law: of electrostatics, 19, 30, 94, 95 for magnetic flux, 18, 28, 29 of magnetostatics, 30, 95 Geometric objects and orientation, 7-15 Geometry, 45 cell-complexes, 46-49 chain boundaries, 54- 55 chains, 52- 54 incidence numbers, 49-52 primary and secondary mesh, 49 Global quantities, 17-18
H Heat transfer equation, 39-44 Higher-order Laue zones (HOLZ), 165, 174
282
INDEX
High-resolution transmission electron microscope (HRTEM), 150-51 Hodge star operator, 97-98
Internal orientation. See Orientation, external and internal Intracolumn interference effects, 171 Inverse limit system, 61-62
K Image processing, quantitative: absence of phase problem, 190-91 deconvolution methods, 192-97 probe reconstruction, 191-92 Imaging: See also under Magnetic domain imaging; under type of difference between coherent and incoherent, 150-52 Imaging of ferromagnetic domain boundaries using PEEM: image calculation formulas, 207-10 image contrast estimation, 225-28 numerical calculation of electron trajectories, 217-20 stray fields, image calculations for, 210-17 stray fields, types of, 207 without restriction of electron beam, 206-22 with restriction of electron beam, 222-28 Incidence numbers, 49-52 Incipient p-cochains, 74 Incoherent imaging: See also Transverse incoherence advantages of, 149-50 difference between coherent and, 150-52 Information limit, 181-90
Kerr effect. See Magnetic domain imaging in UV-PEEM, using Kerr-effect-like contrast Kinematical approximation, TDS and, 175 Kinematical scattering, 171-73
L Laser scanning microscopy (LSM), 252 multiphoton, 256-76 Law of conservation: of electric charge, 28, 29-30 of magnetic flux, 29 Laws, topological. See Topological laws Least squares method, 134 Limit systems, 60-63 inverse, 61-62 Linear magnetic dichroism (LMD), 241 Local quantities, 17-18 Longitudinal coherence, 170 dynamical scattering, 173-74 higher-order Laue zones (HOLZ), 174 kinematical scattering, 171-73 thermal diffuse scattering, 175-81 Lorentz force, 208-10, 240
INDEX
M Magnetic domain imaging in UVPEEM, using Kerr-effectlike contrast: first detection of, 241-42 manifestation in threshold PEEM, 239-41 Magnetic domain imaging in X-PEEM, using magnetic X-ray circular dichroism (MXCD): exchange-coupled systems and probing depth, 236- 38 micropatterned structures and domain walls, 233-36 principle of element-selective, 228-33 Magnetic field, internal orientation and, 23 Magnetic flux, 9 conservation law of, 29 Gauss's law for, 18, 28, 29 Magnetic linear dichroism in angular distribution (MLDAD), 241 Magnetic X-ray circular dichroism (MXCD). See Magnetic domain imaging in X-PEEM, using magnetic X-ray circular dichroism (MXCD) Maxwell-Amp~re's law, 19, 30-31 DSI discretization of, 123-24 FIT of, 129 topological time-stepping, 89, 91-94 Maxwell Grid Equations, 129 Maxwell's equations, 9 Mesh, primary and secondary, 49 Modeling errors, 36 Modeling step, 2
283
Multiphoton LSM, 256-76 adjustment of thickness and resolution, 275-76 benefits of, 256-57 evolution of, 273 excitation, 257-58 future prospects, 276 illumination path, optimizing, 275 laser illumination requirements, 259-62 laser interactions with specimens, 265-73 non-imaging detection optics, 263-65 optical components, selection of, 274-75 optical sectioning by nonlinear excitation, 258- 59 probes, selection of, 273-74 sources for excitation, 262-63 Multiplicative deconvolution, 192-94 Multislice calculations, TDS and, 175-78 Multivectors, 76 weighted, 77-78
N Natural basis, 54, 58 Nel type stray field, 207, 212, 214 N incidence matrices, 51 Nodes, 46, 131 Numerical methods for partial differential equations, 2
O Object function: real-space, 163 residual (ROF), 167-69
284
INDEX
Object function, thin specimen, 161 amplitude object, 162 phase object, 162-64 Objective lens, 149 Ohm's laws, 69-70 Optical transfer function (OTF), 157-58, 182, 191-92, 194 Orientation, compatible or coherent, 52 Orientation, external and internal: current and external, 23 defined, 8 duality of, 13-14 external, for geometric objects in 3-dimensional space, 11-13 in Faraday's law, 9, 10 internal, for geometric objects in 3-dimensional space, 9-10, 12 magnetic field and internal, 23 right-hand rule, 9-10
P p-cell, 46-47 p-dimensional cell, 8, 11, 15, 46 p-dimensional chain with real coefficients, 54 p-dimensional cochain: defined, 57 ordinary versus twisted, 57 p-dimensional differential form, 74 Perpendicular type stray field, 207, 214-17 p-form: defined, 74 ordinary versus twisted, 74 Phase problem, 190-91 Photoemission electron microscope (PEEM), applications, 205-6
Photoemission electron microscope, imaging of ferromagnetic domain boundaries using: See also Magnetic domain imaging in UV-PEEM, using Kerr-effect-like contrast; Magnetic domain imaging in X-PEEM, using magnetic X-ray circular dichroism (MXCD) image calculation formulas, 207-10 image contrast estimation, 225-28 numerical calculation of electron trajectories, 217-20 stray fields, image calculations for, 210-17 stray fields, types of, 207 without restriction of electron beam, 206-22 with restriction of electron beam, 222-28 Physical field problem/theories: alternative methods for, 3, 4 boundary conditions and sources, 37-38 classification of physical quantities, 22-27 constitutive equations, 19-20, 33-37 constitutive relations, 69-72 continuous representations, 72-86 defined, 1-2 discretization strategy, reference, 86-112 equations and physical quantities, 15-22 fields, 55-63 finite difference methods, 112-22
INDEX
finite element methods, 131-40 finite volume methods, 122-31 geometric objects and orientation, 7-15 geometry, 45-55 mathematical structure of, 5-7 structural approach, 38-45 topological laws, 27-33, 63-69 Physical laws, defined, 6 Physical quantities: classification of, 22-27 defined, 15 equations, 18-22 local and global, 17-18 space-time, 24-27 Prime operator, 119 Probability distribution, 195-97 Probe reconstruction, 191-92 p-skeleton, 48 p-vector, 76 Pullback, 75-76
Q q-dimensional faces, proper, 47
R Rayleigh, Lord, 151-52, 157, 161 Residual equations, 132 Resolution: chromatic aberrations, 185-89 source size and ultimate, 189-90 underfocused microscopy, 181-84 Resolution limit, transverse incoherence and, 160-61 Riemann integral, 75-78 Roth's diagrams, 6 Runge-Kutta method, 218
285
S Scanning electron microscope (SEM), differences between STEM and, 148-49 Scanning optical microscope (SOM), 252 Scanning transmission electron microscope (STEM): description of, 148-49 image formation, 153-56 Separation of variables technique, 135 Shape functions: defined, 106- 7 finite element, 132 ordinary versus twisted, 107 Solver errors, 36-37 Space-time domain discretization, 87-89 Space-time objects, 14-15, 24-27 Spread cells, 81-85 Stokes's theorem, 20 Stray fields: image calculations for, 210-17 types of, 207 Structure of a physical theory, defined, 5 Subdomain method, 134 Subtractive deconvolution, 194 Summation by parts formula, 83-84 Support Operator Method (SOM), 118-21
T Thermal diffuse scattering (TDS), 175-81 Bloch wave calculations and, 178-81
286
INDEX
Thermal diffuse scattering (Cont.) kinematical approximation and, 175 multislice calculations and, 175-78 Thin specimen object function, 161 amplitude object, 162 phase object, 162-64 Time-domain: edge element method, 136-39 error-based finite element method, 139-40 finite element method, 134-36 Time-stepping: FDTD formulas, 113, 114-18 topological, 89-96 Tonti classification diagrams, 6-7 for electostatics, 38, 40 of global electromagnetic quantities, 25-27, 59, 68 for heat transfer equation, 39-44 of local electromagnetic quantities, 24 for magnetostatistics, 38, 41 of physical quantities, 14 of space-time, 16 Topological laws, 27-33 coboundary operator and, 64-67 discrete representation for, 63-69 weak form of, 85-86 Topological time-stepping, 89-96 Transformation diagrams, 6
Transmission electron microscope (TEM), high-resolution, 150-51 Transverse incoherence: conditions for, 157-60 dynamical scattering using Bloch waves, 165-70 resolution limit, 160-61 STEM image formation, 153-56 thin specimen object function, 161-64
U Underfocused microscopy, 181-84
V Variational approach, 131 Vector elements, 109
W Weak-phase object approximation (WPOA), 158 Weighted integrals, 75-78 Weighted multivectors, 77-78 Weighted residual technique, 73, 131-32 Whitney functor, 102 Wiener filter, 193, 195
