Principles of Electron Optics: Basic Geometrical Optics

Cover
Principles of Electron Optics,
Volume One: Basic Geometrical Optics
Copyright
Preface to the Second Edition
Preface to the First Edition (Extracts)
Acknowledgments
1 Introduction
	1.1 Organization of the Subject
	1.2 History
Part I: Classical Mechanics
2 Relativistic Kinematics
	2.1 The Lorentz Equation and General Considerations
	2.2 Conservation of Energy
	2.3 The Acceleration Potential
	2.4 Definition of Coordinate Systems
	2.5 Conservation of Axial Angular Momentum
3 Different Forms of Trajectory Equations
	3.1 Parametric Representation in Terms of the Arc-Length
	3.2 Relativistic Proper-Time Representation
	3.3 The Cartesian Representation
	3.4 Scaling Rules
4 Variational Principles
	4.1 The Lagrange Formalism
	4.2 General Rotationally Symmetric Systems
	4.3 The Canonical Formalism
	4.4 The Time-Independent Form of the Variational Principle
	4.5 Static Rotationally Symmetric Systems
5 Hamiltonian Optics
	5.1 Introduction of the Characteristic Function
	5.2 The Hamilton–Jacobi Equation
	5.3 The Analogy With Light Optics
	5.4 The Influence of Vector Potentials
	5.5 Gauge Transformations
	5.6 Poincaré’s Integral Invariant
	5.7 The Problem of Uniqueness
	5.8 Lie Algebra
	5.9 Summary
Part II: Calculation of Static Fields
6 Basic Concepts and Equations
	6.1 General Considerations
	6.2 Field Equations
	6.3 Variational Principles
	6.4 Rotationally Symmetric Fields
	6.5 Planar Fields
7 Series Expansions
	7.1 Azimuthal Fourier Series Expansions
		7.1.1 Scalar Potentials
		7.1.2 Vector Potentials
	7.2 Radial Series Expansions
		7.2.1 Scalar Potentials
		7.2.2 Vector Potentials
		7.2.3 Explicit Representations
	7.3 Rotationally Symmetric Fields
		7.3.1 Electrostatic Fields
		7.3.2 Magnetic Fields
	7.4 Multipole Fields
	7.5 Planar Fields
	7.6 Fourier–Bessel Series Expansions
8 Boundary-Value Problems
	8.1 Boundary-Value Problems in Electrostatics
	8.2 Boundary Conditions in Magnetostatics
	8.3 Examples of Boundary-Value Problems in Magnetostatics
		8.3.1 Devices with Superconducting Yokes
		8.3.2 Conventional Round Magnetic Lenses
		8.3.3 Unconventional Round Magnetic Lenses
		8.3.4 Toroidal Magnetic Deflection Systems
9 Integral Equations
	9.1 Integral Equations for Scalar Potentials
		9.1.1 General Theory
		9.1.2 Dirichlet Problems
		9.1.3 Neumann Problems
	9.2 Problems with Interface Conditions
	9.3 Reduction of the Dimensions
		9.3.1 Dirichlet Problems
		9.3.2 Interface Conditions
		9.3.3 Planar Fields
	9.4 Important Special Cases
		9.4.1 Rotationally Symmetric Scalar Potentials
		9.4.2 Rotationally Symmetric Vector Potentials
		9.4.3 Unconventional Magnetic Lenses
		9.4.4 Magnetic Deflection Coils
		9.4.5 Multipole Systems
		9.4.6 Small Perturbations of the Rotational Symmetry
	9.5 Résumé
10 The Boundary-Element Method
	10.1 Evaluation of the Fourier Integral Kernels
		10.1.1 Introduction of Moduli
		10.1.2 Radial Series Expansions
		10.1.3 Recurrence Relations
		10.1.4 Analytic Differentiation
	10.2 Numerical Solution of One-Dimensional Integral Equations
		10.2.1 Conventional Solution Techniques
		10.2.2 The Charge Simulation Method
		10.2.3 Combination with Interpolation Kernels
			10.2.3.1 General formalism
			10.2.3.2 Marginal positions
			10.2.3.3 General properties
			10.2.3.4 Solution of integral equations
			10.2.3.5 Application to field calculations
		10.2.4 Evaluation of Improper Integrals
	10.3 Superposition of Aperture Fields
		10.3.1 Electric Field of a Single Aperture
		10.3.2 Superposition Procedure
		10.3.3 Combination with the BEM
		10.3.4 Extrapolation of the Number of Segments
	10.4 Three-Dimensional Dirichlet Problems
	10.5 Examples of Applications of the Boundary-Element Method
11 The Finite-Difference Method (FDM)
	11.1 The Choice of Grid
	11.2 The Taylor Series Method
	11.3 The Integration Method
	11.4 Nine-Point Formulae
	11.5 The Finite-Difference Method in Three Dimensions
	11.6 Other Aspects of the Method
		11.6.1 Expanding Spherical-Mesh Grid
		11.6.2 Extrapolation on Multiple Grids
		11.6.3 Combination with the BEM
	11.7 Iterative Solution Techniques
12 The Finite-Element Method (FEM)
	12.1 Formulation for Round Magnetic Lenses
	12.2 Formulation for Self-adjoint Elliptic Equations
	12.3 Solution of the Finite-Element Equations
	12.4 Improvement of the Finite-Element Method
		12.4.1 Introduction
		12.4.2 Alternative Formulations
		12.4.3 First- and Second-Order Finite-Element Methods (FOFEM and SOFEM)
	12.5 Comparison and Combination of Different Methods
	12.6 Deflection Units and Multipoles
	12.7 Related Work
13 Field-Interpolation Techniques
	13.1 One-Dimensional Differentiation and Interpolation
		13.1.1 Hermite Interpolation
		13.1.2 Cubic Splines
		13.1.3 Differentiation Using Difference Schemes
		13.1.4 Evaluation of Radial Series Expansions
	13.2 Two-Dimensional Interpolation
		13.2.1 Hermite Interpolation
		13.2.2 The Use of Derivatives of Higher Order
	13.3 Interpolation and the Finite-Element Method
Part III: The Paraxial Approximation
14 Introduction to Paraxial Equations
15 Systems with an Axis of Rotational Symmetry
	15.1 Derivation of the Paraxial Ray Equations from the General Ray Equations
		15.1.1 Physical Significance of the Coordinate Rotation
	15.2 Variational Derivation of the Paraxial Equations
	15.3 Forms of the Paraxial Equations and General Properties of their Solutions
		15.3.1 Reduced Coordinates
		15.3.2 Stigmatic Image Formation
		15.3.3 The Wronskian
	15.4 The Abbe Sine Condition and Herschel’s Condition
	15.5 Some Other Transformations
16 Gaussian Optics of Rotationally Symmetric Systems: Asymptotic Image Formation
	16.1 Real and Asymptotic Image Formation
	16.2 Asymptotic Cardinal Elements and Transfer Matrices
	16.3 Gaussian Optics as a Projective Transformation (Collineation)
	16.4 Use of the Angle Characteristic to Establish the Gaussian Optical Quantities
	16.5 The Existence of Asymptotes
17 Gaussian Optics of Rotationally Symmetric Systems: Real Cardinal Elements
	17.1 Real Cardinal Elements for High Magnification and High Demagnification
	17.2 Osculating Cardinal Elements
	17.3 Inversion of the Principal Planes
	17.4 Approximate Formulae for the Cardinal Elements: The Thin-Lens Approximation and the Weak-Lens Approximation
		Magnetic Lenses
		Electrostatic Lenses
18 Electron Mirrors
	18.1 Introduction
	18.2 The Modified Temporal Representation
	18.3 The Cartesian Representation
	18.4 A Quadratic Transformation
19 Quadrupole Lenses
	19.1 Paraxial Equations for Quadrupoles
	19.2 Transaxial Lenses
20 Cylindrical Lenses
Part IV: Aberrations
21 Introduction to Aberration Theory
22 Perturbation Theory: General Formalism
23 The Relation Between Permitted Types of Aberration and System Symmetry
	23.1 Introduction
	23.2 N=1
		23.2.1 N=1. Systems with a Plane of Symmetry
	23.3 N=2
		23.3.1 N=2. Systems Possessing a Plane of Symmetry
	23.4 N=3
	23.5 N=4
	23.6 N=5 and 6
	23.7 Systems with an Axis of Rotational Symmetry
	23.8 Note on the Classification of Aberrations
		23.8.1 Terms Independent of xo, yo (p=q=0): Aperture Aberrations
		23.8.2 Terms Independent of xa, ya (r=s=0): Distortions
		23.8.3 Intermediate Terms
		23.8.4 Phase Shifts
		23.8.5 Parasitic Aberrations
24 The Geometrical Aberrations of Round Lenses
	24.1 Introduction
	24.2 Derivation of the Real Aberration Coefficients
		24.2.1 The Trajectory Method
		24.2.2 The Eikonal Method
	24.3 Spherical Aberration (Terms in xa and ya only)
		24.3.1 Electrostatic case (B=0, φ ≠ const)
			General Relativistic Expression
			General Nonrelativistic Expression
		24.3.2 Magnetic case (φ=const, B≠0)
			General Relativistic Case
		24.3.3 Scherzer’s Theorem
		24.3.4 Thin-Lens Approximation
	24.4 Coma (Terms Linear in xo, yo)
		24.4.1 Thin-Lens Formulae
	24.5 Astigmatism and Field Curvature (Terms Linear in xa, ya)
		24.5.1 Thin-Lens Formulae
	24.6 Distortion (Terms in xo and yo only)
		24.6.1 Thin-Lens Formulae
	24.7 The Variation of the Aberration Coefficients with Aperture Position
	24.8 Reduced Coordinates4
	24.9 Seman’s Transformation of the Characteristic Function
	24.10 Fifth-Order Aberrations
		24.10.1 Isotropic Aberration Coefficients
		24.10.2 Anisotropic Aberration Coefficients
25 Asymptotic Aberration Coefficients
	25.1 Spherical Aberration
	25.2 Coma
	25.3 Astigmatism and Field Curvature
	25.4 Distortion
	25.5 Aberration Matrices and the Integrals ij
	25.6 Dependence on Object Position or Magnification
	25.7 Dependence on Aperture Position
	25.8 Thin-Lens Approximations
26 Chromatic Aberrations
	26.1 Real Chromatic Aberrations
	26.2 Asymptotic Chromatic Aberrations
	26.3 Higher Order Chromatic Aberration Coefficients
		26.3.1 Third-Order (Fourth-Rank) Aberrations
			26.3.1.1 Isotropic Aberrations
			26.3.1.2 Anisotropic Aberrations
			26.3.1.3 Definitions
		26.3.2 Third-Rank Aberrations
27 Aberration Matrices and the Aberrations of Lens Combinations
28 The Aberrations of Mirrors and Cathode Lenses
	28.1 The Modified Temporal Theory
	28.2 The Cartesian Theory
	28.3 Devices with Curved Cathodes
	28.4 Practical Mirror Studies
29 The Aberrations of Quadrupole Lenses and Octopoles
	29.1 Introduction
	29.2 Geometrical Aberration Coefficients
	29.3 Aperture Aberrations
	29.4 Chromatic Aberrations
	29.5 Quadrupole Multiplets
30 The Aberrations of Cylindrical Lenses
31 Parasitic Aberrations
	31.1 Small Deviations from Rotational Symmetry; Axial Astigmatism
	31.2 Classification of the Parasitic Aberrations
	31.3 Numerical Determination of Parasitic Aberrations
		31.3.1 Introduction
		31.3.2 Use of the Finite-Difference Method
		31.3.3 Use of the Finite-Element Method
	31.4 The Isoplanatic Approximation
	31.5 Stigmators
		31.5.1 Necessary Simplifications
		31.5.2 The Wave Aberration
		31.5.3 The Deflection of Trajectories
	31.6 Advanced Theory
		31.6.1 Introduction
		31.6.2 Notation
		31.6.3 Further Analysis of the Aberrations, Classified by Order
	31.7 The Uhlemann Effect
Part V: Deflection Systems
32 Paraxial Properties of Deflection Systems
	32.1 Introduction
	32.2 The Paraxial Optics of Deflection Systems
		32.2.1 The General Paraxial Equations
		32.2.2 Ideal Deflection
		32.2.3 The Dependence on the Electrical Input Signals
		32.2.4 Rotation-Invariant Systems
33 The Aberrations of Deflection Systems
	33.1 Pure Deflection Systems
		33.1.1 Two Different Symmetries
		33.1.2 Fourfold Symmetry
		33.1.3 General Considerations
	33.2 Deflection Systems with Magnetic Lenses
		33.2.1 Geometric Aberrations
		33.2.2 Chromatic Aberrations
	33.3 Detailed Aberration Analyses
Part VI: Computer-Aided Electron Optics
34 Numerical Calculation of Trajectories, Paraxial Properties and Aberrations
	34.1 Introduction
	34.2 Numerical Solution of Ordinary Differential Equations
		34.2.1 The Fox–Goodwin–Numerov Method
		34.2.2 The Runge–Kutta Method
		34.2.3 The Predictor–Corrector Method
		34.2.4 Special Considerations
	34.3 Standard Applications in Electron Optics
		34.3.1 Initial-Value Problems
		34.3.2 Boundary-Value Problems
	34.4 Differential Equations for the Aberrations
		34.4.1 Electrostatic Systems With a Straight Optic Axis
		34.4.2 Separation in Arbitrary Systems
		34.4.3 Chromatic Shifts
	34.5 Least-Squares-Fit Methods in Electron Optics
		34.5.1 General Complex Formulation
		34.5.2 The Determination of Deflection Aberrations
		34.5.3 Some Other Examples
	34.6 Determination and Evaluation of Aberration Discs
		34.6.1 Fourier Analysis of the Aberrations
		34.6.2 Some Practical Aspects
		34.6.3 Integral Properties of Aberration Discs
	34.7 Optimization Procedures
		34.7.1 The Defect Function
		34.7.2 The Optimization of Axial Distributions
		34.7.3 The Damped Least-Squares Method
	34.8 Differential Algebra
		34.8.1 Introduction
		34.8.2 Definition of Differential Algebras
		34.8.3 Calculation of Aberration Coefficients
	34.9 The Use of Computer Algebra Languages
		34.9.1 Introduction
		34.9.2 Computer Algebra, Its Role in Electron Optics
		34.9.3 Practical Examples
Notes and References
	Preface and Chapter 1
	Part I, Chapters 2–5
	Part II, Chapters 6–13
	Part III, Chapters 14–20
	Part IV, Chapters 21–31
	Part V, Chapters 32 and 33
	Part VI, Chapter 34
	Conference Proceedings
Index
Back Cover