# Advances in Imaging and Electron Physics

**Selected problems of computational charged particle optics**

**Series Editor:**

- Peter Hawkes, CEMES/Laboratoire d'Optique Electronique du Centre National de la Recherche Scientifique, Toulouse, France

**By**- Dmitry Greenfield
- Mikhael Monastyrskii

*Advances in Imaging and Electron Physics* merges two long-running serials--*Advances in Electronics and Electron Physics* and *Advances in Optical and Electron Microscopy.* This series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. This monograph summarizes the authors' knowledge and experience acquired over many years in their work on computational charged particle optics. Its main message is that even in this era of powerful computers with a multitude of general-purpose and problem-oriented programs, asymptotic analysis based on perturbation theory remains one of the most effective tools to penetrate deeply into the essence of the problem in question.

View full description### Audience

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

### Book information

- Published: December 2008
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-374717-4

### Table of Contents

Chapter 1. Integral equations method in electrostatics1.1. Statement of the problem1.2. Boundary surface approximation1.3. Surface charge density approximation 1.4. Interface boundary conditions for dielectric materials1.5 Reducing the integral equations to the finite-dimensional linear equations system 1.6. Accuracy benchmarks for numerical solving the 3D electrostatic problems1.7. More complicated examples of 3D field simulation1.8 The cases of planar and axial symmetries 1.9 Calculation of potential and its derivatives near the boundary1.10. Acceleration of field calculation. Finite-difference meshes and calculation domain decomposition1.11. Microscopic and averaged fields of periodic structuresChapter 2. Surface charge singularities near irregular surface points2.1. Two-faced conductive wedge in vacuum 2.2. Two-faced conductive wedge in the presence of dielectrics2.3. The transfer matrix method2.4. The case of pure dielectric vertex2.5. Upper boundaries for the singularity index in 2D case2.6. Variational approach to the spectral problem2.7. Three-dimensional corners2.8. Variational method in the case of dielectrics2.9. Reduction to the 2D case2.10. On-rib singularities near three-dimensional corner2.11. The cases allowing separation of variables2.12. Numerical solution of the Beltrami-Laplace spectral problem2.13. Cubical and prism cornersChapter 3. Geometry perturbations3.1. Integral variational equations and conjugate integral equation for the Green function3.2. 3D perturbations in axisymmetric systems 3.3 Some examples of 3D perturbations in axisymmetric systems3.4. 3D perturbations in planar systems3.5. Locally strong 3D perturbations in axisymmetric systems3.6. 3D fringe fields in planar systemsChapter 4. Some aspects of magnetic field simulation4.1. Vector and scalar potential approaches 4.2. Direct integration over the current contours4.3. The current contours in the presence of materials with constant permeability4.4. Variational principle in three-dimensional, planar, and axisymmetric cases4.5. Finite-element modeling of magnetic systems with saturable materials4.6. Second-order FEM and the use of curvilinear elements4.7. Magnetic superelements4.8. The boundary element approach in magnetostatics4.9. Hybrid computational methodsChapter 5. Aberration approach and the tau-variation technique5.1. A brief excursion to the history of aberration theory5.2. The essence of the tau-variation technique 5.3. The tau-variation equations in tensor form 5.4. Arrival time variations and contact transformation5.5. Jump condition for aberration coefficients 5.6. Multiple principal trajectories approach 5.7. Tolerance analysis using the aberration theory5.8. Tracking technique5.9. Charged particle scatteringChapter 6. Space charge in charged particle bunches6.1. Self-consistent simulation of thermionic electron guns6.2. Cold-cathode approximation: semi-analytical approach6.3. Coulomb field in short bunches. The technique of tree-type pre-ordering6.4. Exclusion of the external field in space charge problems6.5. Some examples of ion beam simulationChapter 7. General properties of emission-imaging systems7.1. Charged particle density transformations and electron image7.2. Spatial/temporal spread function. Isoplanatism condition7.3. Modulation and phase transfer functions (MTF and PTF). Spatial and temporal resolutionChapter 8. Static and time-analyzing image tubes with axial symmetry8.1. Spatial aberrations of the electron image formed by electrostatic systems8.2. Temporal aberrations in streak image tubes8.3. High-frequency asymptotics of OTF in image tubes8.4. Examples of the spread functions and OTF in the image tubes8.5. The boundary-layer effect in cathode lenses and electron mirrorsChapter 9. Spatial and temporal focusing of photoelectron bunches in time-dependent electric fields9.1. Two different jobs that ultrashort electron bunches can do9.2. The master equation of first-order temporal focusing9.3. Moving potential well as a simple example of temporal focusing9.4. Thin temporal lens approximation9.5. Second-order aberrations and quantum-mechanical limitations9.6. Approximate estimation of the space charge effects contribution9.7. Simulation of a photoelectron gun with time-dependent electric field and some experimental resultsAppendicesAppendix 1. Some Gauss quadrature formulasAppendix 2. Numerical integration of the Green functions with Coulomb singularities in the coincidence limitAppendix 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 partAppendix 5. Some general properties of linear systemsAppendix 6. The probability density transformationsAppendix 7. The multidimensional stationary phase methodReferences