# Advances in Imaging and Electron Physics, Volume 155

## 1st Edition

### Selected Problems of Computational Charged Particle Optics

**Authors:**Dmitry Greenfield Mikhael Monastyrskii

**Serial Editors:**Peter Hawkes

**Hardcover ISBN:**9780123747174

**eBook ISBN:**9780080879697

**Imprint:**Academic Press

**Published Date:**29th December 2008

**Page Count:**364

**View all volumes in this series:**Advances in Imaging and Electron Physics

## Table of Contents

Chapter 1. Integral equations method in electrostatics
1.1. Statement of the problem
1.2. Boundary surface approximation
1.3. Surface charge density approximation

1.4. Interface boundary conditions for dielectric materials
1.5 Reducing the integral equations to the finite-dimensional linear equations system

1.6. Accuracy benchmarks for numerical solving the 3D electrostatic problems
1.7. More complicated examples of 3D field simulation
1.8 The cases of planar and axial symmetries

1.9 Calculation of potential and its derivatives near the boundary
1.10. Acceleration of field calculation. Finite-difference meshes and calculation domain decomposition
1.11. Microscopic and averaged fields of periodic structures

Chapter 2. Surface charge singularities near irregular surface points
2.1. Two-faced conductive wedge in vacuum

2.2. Two-faced conductive wedge in the presence of dielectrics
2.3. The transfer matrix method
2.4. The case of pure dielectric vertex
2.5. Upper boundaries for the singularity index in 2D case
2.6. Variational approach to the spectral problem
2.7. Three-dimensional corners
2.8. Variational method in the case of dielectrics
2.9. Reduction to the 2D case
2.10. On-rib singularities near three-dimensional corner
2.11. The cases allowing separation of variables
2.12. Numerical solution of the Beltrami-Laplace spectral problem
2.13. Cubical and prism corners

Chapter 3. Geometry perturbations
3.1. Integral variational equations and conjugate integral equation for the Green function
3.2. 3D perturbations in axisymmetric systems

3.3 Some examples of 3D perturbations in axisymmetric systems
3.4. 3D perturbations in planar systems
3.5. Locally strong 3D perturbations in axisymmetric systems
3.6. 3D fringe fields in planar systems

Chapter 4. Some aspects of magnetic field simulation
4.1. Vector and scalar potential approaches

4.2. Direct integration over the current contours
4.3. The current contours in the presence of materials with constant permeability
4.4. Variational principle in three-dimensional, planar, and axisymmetric cases
4.5. Finite-element modeling of magnetic systems with saturable materials
4.6. Second-order FEM and the use of curvilinear elements
4.7. Magnetic superelements
4.8. The boundary element approach in magnetostatics
4.9. Hybrid computational methods

Chapter 5. Aberration approach and the tau-variation technique
5.1. A brief excursion to the history of aberration theory
5.2. The essence of the tau-variation technique

5.3. The tau-variation equations in tensor form

5.4. Arrival time variations and contact transformation
5.5. Jump condition for aberration coefficients

5.6. Multiple principal trajectories approach

5.7. Tolerance analysis using the aberration theory
5.8. Tracking technique
5.9. Charged particle scattering

Chapter 6. Space charge in charged particle bunches 6.1. Self-consistent simulation of thermionic electron guns 6.2. Cold-cathode approximation: semi-analytical approach 6.3. Coulomb field in short bunches. The technique of tree-type pre-ordering 6.4. Exclusion of the external field in space charge problems 6.5. Some examples of ion beam simulation

Chapter 7. General properties of emission-imaging systems 7.1. Charged particle density transformations and electron image 7.2. Spatial/temporal spread function. Isoplanatism condition 7.3. Modulation and phase transfer functions (MTF and PTF). Spatial and temporal resolution

Chapter 8. Static and time-analyzing image tubes with axial symmetry 8.1. Spatial aberrations of the electron image formed by electrostatic systems 8.2. Temporal aberrations in streak image tubes 8.3. High-frequency asymptotics of OTF in image tubes 8.4. Examples of the spread functions and OTF in the image tubes 8.5. The boundary-layer effect in cathode lenses and electron mirrors

Chapter 9. Spatial and temporal focusing of photoelectron bunches in time-dependent electric fields 9.1. Two different jobs that ultrashort electron bunches can do 9.2. The master equation of first-order temporal focusing 9.3. Moving potential well as a simple example of temporal focusing 9.4. Thin temporal lens approximation 9.5. Second-order aberrations and quantum-mechanical limitations 9.6. Approximate estimation of the space charge effects contribution 9.7. Simulation of a photoelectron gun with time-dependent electric field and some experimental results

Appendices Appendix 1. Some Gauss quadrature formulas Appendix 2. Numerical integration of the Green functions with Coulomb singularities in the coincidence limit Appendix 3. First variation of a functional upon the equality-type operator constraints (R.P. Fedorenko’ variational scheme) Appendix 4. Jump condition for variations of the ordinary differential equations with non-smooth right part Appendix 5. Some general properties of linear systems Appendix 6. The probability density transformations Appendix 7. The multidimensional stationary phase method

References

## Description

Chapter 1. Integral equations method in electrostatics
1.1. Statement of the problem
1.2. Boundary surface approximation
1.3. Surface charge density approximation

1.4. Interface boundary conditions for dielectric materials
1.5 Reducing the integral equations to the finite-dimensional linear equations system

1.6. Accuracy benchmarks for numerical solving the 3D electrostatic problems
1.7. More complicated examples of 3D field simulation
1.8 The cases of planar and axial symmetries

1.9 Calculation of potential and its derivatives near the boundary
1.10. Acceleration of field calculation. Finite-difference meshes and calculation domain decomposition
1.11. Microscopic and averaged fields of periodic structures

Chapter 2. Surface charge singularities near irregular surface points
2.1. Two-faced conductive wedge in vacuum

2.2. Two-faced conductive wedge in the presence of dielectrics
2.3. The transfer matrix method
2.4. The case of pure dielectric vertex
2.5. Upper boundaries for the singularity index in 2D case
2.6. Variational approach to the spectral problem
2.7. Three-dimensional corners
2.8. Variational method in the case of dielectrics
2.9. Reduction to the 2D case
2.10. On-rib singularities near three-dimensional corner
2.11. The cases allowing separation of variables
2.12. Numerical solution of the Beltrami-Laplace spectral problem
2.13. Cubical and prism corners

Chapter 3. Geometry perturbations
3.1. Integral variational equations and conjugate integral equation for the Green function
3.2. 3D perturbations in axisymmetric systems

3.3 Some examples of 3D perturbations in axisymmetric systems
3.4. 3D perturbations in planar systems
3.5. Locally strong 3D perturbations in axisymmetric systems
3.6. 3D fringe fields in planar systems

Chapter 4. Some aspects of magnetic field simulation
4.1. Vector and scalar potential approaches

4.2. Direct integration over the current contours
4.3. The current contours in the presence of materials with constant permeability
4.4. Variational principle in three-dimensional, planar, and axisymmetric cases
4.5. Finite-element modeling of magnetic systems with saturable materials
4.6. Second-order FEM and the use of curvilinear elements
4.7. Magnetic superelements
4.8. The boundary element approach in magnetostatics
4.9. Hybrid computational methods

Chapter 5. Aberration approach and the tau-variation technique
5.1. A brief excursion to the history of aberration theory
5.2. The essence of the tau-variation technique

5.3. The tau-variation equations in tensor form

5.4. Arrival time variations and contact transformation
5.5. Jump condition for aberration coefficients

5.6. Multiple principal trajectories approach

5.7. Tolerance analysis using the aberration theory
5.8. Tracking technique
5.9. Charged particle scattering

Chapter 6. Space charge in charged particle bunches 6.1. Self-consistent simulation of thermionic electron guns 6.2. Cold-cathode approximation: semi-analytical approach 6.3. Coulomb field in short bunches. The technique of tree-type pre-ordering 6.4. Exclusion of the external field in space charge problems 6.5. Some examples of ion beam simulation

Chapter 7. General properties of emission-imaging systems 7.1. Charged particle density transformations and electron image 7.2. Spatial/temporal spread function. Isoplanatism condition 7.3. Modulation and phase transfer functions (MTF and PTF). Spatial and temporal resolution

Chapter 8. Static and time-analyzing image tubes with axial symmetry 8.1. Spatial aberrations of the electron image formed by electrostatic systems 8.2. Temporal aberrations in streak image tubes 8.3. High-frequency asymptotics of OTF in image tubes 8.4. Examples of the spread functions and OTF in the image tubes 8.5. The boundary-layer effect in cathode lenses and electron mirrors

Chapter 9. Spatial and temporal focusing of photoelectron bunches in time-dependent electric fields 9.1. Two different jobs that ultrashort electron bunches can do 9.2. The master equation of first-order temporal focusing 9.3. Moving potential well as a simple example of temporal focusing 9.4. Thin temporal lens approximation 9.5. Second-order aberrations and quantum-mechanical limitations 9.6. Approximate estimation of the space charge effects contribution 9.7. Simulation of a photoelectron gun with time-dependent electric field and some experimental results

Appendices Appendix 1. Some Gauss quadrature formulas Appendix 2. Numerical integration of the Green functions with Coulomb singularities in the coincidence limit Appendix 3. First variation of a functional upon the equality-type operator constraints (R.P. Fedorenko’ variational scheme) Appendix 4. Jump condition for variations of the ordinary differential equations with non-smooth right part Appendix 5. Some general properties of linear systems Appendix 6. The probability density transformations Appendix 7. The multidimensional stationary phase method

References

## Readership

Physicists, electrical engineers and applied mathematicians in all branches of image processing and microscopy as well as electron physics in general

## Details

- No. of pages:
- 364

- Language:
- English

- Copyright:
- © Academic Press 2009

- Published:
- 29th December 2008

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080879697

- Hardcover ISBN:
- 9780123747174

## About the Authors

### Dmitry Greenfield Author

### Mikhael Monastyrskii Author

## About the Serial Editors

### Peter Hawkes Serial Editor

Peter Hawkes graduated from the University of Cambridge and subsequently obtained his PhD in the Electron Microscopy Section of the Cavendish Laboratory. He remained there for several years, working on electron optics and digital image processing before taking up a research position in the CNRS Laboratory of Electron Optics (now CEMES-CNRS) in Toulouse, of which he was Director in 1987. During the Cambridge years, he was a Research Fellow of Peterhouse and a Senior Research fellow of Churchill College. He has published extensively, both books and scientific journal articles, and is a member of the editorial boards of Ultramicroscopy and the Journal of Microscopy. He was the founder-president of the European Microscopy Society, CNRS Silver Medallist in 1983 and is a Fellow of the Optical Society of America and of the Microscopy Society of America (Distinguished Scientist, Physics, 2015), Fellow of the Royal Microscopical Society and Honorary Member of the French Microscopy Society. In 1982, he was awarded the ScD degree by the University of Cambridge.

In 1982, he took over editorship of the Advances in Electronics & Electron Physics (now Advances in Imaging & Electron Physics) from Claire Marton (widow of the first editor, Bill Marton) and followed Marton's example in maintaining a wide range of subject matter. He added mathematical morphology to the topics regularly covered; Jean Serra and Gerhard Ritter are among those who have contributed.

In 1980, he joined Professor Wollnik (Giessen University) and Karl Brown (SLAC) in organising the first international conference on charged-particle optics, designed to bring together opticians from the worlds of electron optics, accelerator optics and spectrometer optics. This was so successful that similar meetings have been held at four-year intervals from 1986 to the present day. Peter Hawkes organised the 1990 meeting in Toulouse and has been a member of the organising committee of all the meetings. He has also participated in the organization of other microscopy-related congresses, notably EMAG in the UK and some of the

He is very interested in the history of optics and microscopy, and recently wrote long historical articles on the correction of electron lens aberrations, the first based on a lecture delivered at a meeting of the Royal Society. He likewise sponsored biographical articles for the Advances on such major figures as Ernst Ruska (Nobel Prize 1986), Helmut Ruska, Bodo von Borries, Jan Le Poole and Dennis Gabor (Nobel Prize, 1971). Two substantial volumes of the series were devoted to 'The Beginnings of Electron Microscopy' and 'The Growth of Electron Microscopy'. and others have covered 'Cold Field Emission Scanning Transmission Electron Microscopy' and 'Aberration-corrected Electron Microscopy', with contributions by all the main personalities of the subject.

### Affiliations and Expertise

Laboratoire d'Optique Electronique du Centre National de la Recherche Scientifique (CEMES), Toulouse, France