Principles of Electron Optics, Volume 4 Advanced Wave Optics

• Main Authors: Hawkes, Peter W, Kasper, Erwin
• Format: eBook
• Language: English
• Published: Chantilly Elsevier Science & Technology 2022; Academic Press
• Edition: 2
• Subjects: Electron optics

Principles of Electron Optics: Second Edition, Advanced Wave Optics provides a self-contained, modern accounting of electron optical phenomena with the Dirac or Schrödinger equation as a starting point. Knowledge of this branch of the subject is essential to understanding electron propagation in ele...

Table of Contents:

• 78.11 The Propagation of Coherence Functions -- 78.11.1 Propagation of the Mutual Intensity in Free Space -- 78.11.2 Propagation of the Mutual Intensity Through a Lens System -- 78.11.3 Propagation of the Cross-Spectral Density and Brightness Through a Lens System -- 78.11.4 Introduction of a Specimen -- 78.12 Coherence and Illumination -- 78.13 Degeneracy and Brightness -- 78.14 Further Reading -- 79 Wigner Optics -- 79.1 Introduction -- 79.2 Image Formation Expressed in Terms of the Wigner Function -- 79.2.1 Source Properties -- 79.2.2 Effect of an Aperture -- 79.2.3 Passage through an Electron Lens -- 79.3 Holography -- 79.3.1 Propagation of the Density Matrix through a Conventional Transmission Electron Microscope -- 79.3.2 Propagation of the Density Matrix through an Electron Microscope Fitted with a Biprism -- 79.3.3 Related Holographic Techniques -- 79.4 Further Reading -- XVII. Vortex Studies, the Quantum Electron Microscope -- 80 Orbital Angular Momentum, Vortex Beams and the Quantum Electron Microscope -- 80.1 Introduction -- 80.2 Vortex Beams -- 80.2.1 Properties -- 80.2.2 Knots and Links -- 80.3 Interaction With Magnetic Fields -- 80.4 Production of Vortex Beams -- 80.4.1 Phase Plates -- 80.4.2 Holograms -- 80.4.3 Aberration Correctors as Vortex Generators -- 80.4.4 Mode Conversion -- 80.4.5 A Mirror-Based Method -- 80.4.6 Vortex Interferometry -- 80.4.7 Structured Illumination -- 80.4.8 Concluding Note -- 80.5 Measurement of Topological Charge -- 80.5.1 Diffraction -- 80.5.2 Stigmators -- 80.5.3 Azimuthal to Cartesian Mapping -- 80.5.4 Induced Currents -- 80.6 Interactions Between Vortex Beams and Specimens -- 80.7 Lensless Fourier Transform Holography -- 80.8 Further Reading -- 80.9 The Quantum Electron Microscope -- Appendix -- Appendix: Corrections and additions to volumes 1, 2 and 3 -- A1 Volume 1 -- Additions -- A2 Volume 2
• Front Cover -- Principles of Electron Optics -- Copyright Page -- Dedication -- Contents -- Preface to the Second Edition -- Preface to the First Edition -- XIV. Electron-Specimen Interactions -- 69 Electron-Specimen Interactions -- 69.1 Introduction -- 69.2 Electron Interactions in Amorphous Specimens -- 69.2.1 Definition of the Elastic Cross-Sections -- 69.2.2 The First-Order Born Approximation for Elastic Scattering -- 69.2.3 The High-Energy Approximation -- 69.2.4 Partial Wave Analysis -- 69.2.4.1 Remaining Problems -- 69.2.5 Inelastic Electron Scattering -- 69.2.5.1 General Properties of Inelastic Scattering -- 69.2.5.2 Further Remarks -- 69.2.6 Plural and Multiple Electron Scattering -- 69.2.7 The Scattering Contrast -- 69.3 Electron Interactions in Crystalline Specimens -- 69.3.1 Introduction -- 69.3.2 Fundamentals of Crystallography -- 69.3.3 The Periodic Potential -- 69.3.4 Kinematic Theory of Electron Scattering -- 69.3.4.1 Geometrical Rules and Scattering -- 69.3.4.2 Intensity Formulae for Diffraction Peaks -- 69.3.4.3 Energy Spread of the Illumination -- 69.3.5 General Formulation of the Dynamical Theory -- 69.3.5.1 The Method of Oscillating Amplitudes -- 69.3.5.2 Formulation as an Eigenvalue Problem -- 69.3.5.3 The Equivalence of the Two Methods -- 69.3.5.4 Absorption -- 69.3.6 The Two-Beam Case -- 69.3.6.1 General Calculations -- 69.3.6.2 Interpretation of the Results -- 69.3.7 Applications and Extensions of the Dynamical Theory -- 69.3.7.1 Nearly Perfect Crystals -- 69.3.7.2 Imperfect Crystals -- 69.4 Simulation and Structure Retrieval -- 69.4.1 Introduction -- 69.4.2 Simulation -- 69.4.3 Reconstruction -- 69.5 Multislice Electron Optics -- XV. Digital Image Processing -- 70 Introduction -- 70.1 Organization of the Subject -- 70.2 Image Algebra -- 70.2.1 Introduction -- 70.2.2 Images and Templates -- 70.2.3 Operations
• 75.4 Three-Dimensional Reconstruction in Materials Science -- 75.5 Deep Learning, Machine Learning -- 75.5.1 Introduction and Principles -- 75.5.2 Noise -- 75.5.3 Segmentation -- 75.5.4 Labelling -- 75.5.5 Sparsity -- 75.5.6 Exit-Wave Reconstruction -- 75.5.7 Tomography -- 75.5.8 General Studies -- 75.6 Concluding Remarks -- 75.7 Further Reading -- 76 Image Analysis -- 76.1 Introduction -- 76.2 Digital Geometry -- 76.2.1 Neighbours -- 76.2.2 Distance -- 76.2.3 Connectedness -- 76.2.4 Border -- 76.2.5 Simplicity -- 76.3 Segmentation and Feature Extraction -- 76.3.1 Segmentation -- 76.3.2 Feature Extraction -- 76.3.3 Measurement -- 76.4 Classification -- 76.5 Description -- 76.6 Further Reading -- 77 Microscope Parameter Measurement and Instrument Control -- 77.1 Introduction -- 77.2 Measurement of Microscope Operating Parameters -- 77.2.1 The Transmission Electron Microscope -- 77.2.2 The Scanning Transmission Electron Microscope -- 77.2.3 Aberration Measurement for Corrected Optics -- 77.2.3.1 The Transmission Electron Microscope -- 77.2.3.2 The Scanning Transmission Electron Microscope -- 77.2.4 Aberration Determination Using Crystalline Materials -- 77.3 Control -- XVI. Coherence, Brightness and Spectral Functions -- 78 Coherence and the Brightness Functions -- 78.1 Introduction -- 78.2 Coherence -- 78.2.1 Definitions -- 78.2.2 Spectral Functions -- 78.3 Radiometry -- 78.4 The Brightness of Partially Coherent Sources -- 78.5 Consequences for the van Cittert-Zernike Theorem -- 78.6 Eigenfunction Expansions of the Coherence Functions -- 78.6.1 The Expansions -- 78.6.2 A New Set of Brightness Formulae -- 78.7 The Quasi-homogeneous Source -- 78.8 Brightness, Coherence and Quasi-homogeneity -- 78.9 Temporal and Spatial Coherence -- 78.10 Related Work -- 78.10.1 Operator Formalism -- 78.10.2 Use of Wigner and Ambiguity Functions
• Omissions and additions
• 74.3.2 The Gerchberg-Saxton Algorithm -- 74.3.3 The Multiple-Image Algorithm -- 74.3.4 Bright-Field/Dark-Field Subtraction -- 74.3.5 Direct Methods -- 74.3.6 Modulation of the Incident Beam -- 74.3.7 One Image and Its Derivative with Respect to Defocus -- 74.3.8 Closely Spaced Images: The Transport-of-Intensity Equation -- 74.3.8.1 Introduction -- 74.3.8.2 Elementary Derivation of the Transport-of-Intensity Equation -- 74.3.8.3 Use of Schrödinger's Equation -- 74.3.8.4 The Notion of Phase -- 74.3.8.5 Further Developments -- 74.3.8.6 Aberration Space, Aberration Coefficients as Variables -- 74.3.9 Related Problems -- 74.4 Analyticity -- 74.4.1 Introduction -- 74.4.2 Analytic Continuation of Wavefunctions -- 74.4.3 Use of Half-Plane Apertures -- 74.4.4 Logarithmic Hilbert Transform Pairs -- 74.4.5 Uniqueness in One and Two Dimensions -- 74.4.5.1 Continuous Form -- 74.4.5.2 Discrete Form -- 74.4.5.3 One-Dimensional Case -- 74.4.5.4 Two- or Higher-Dimensional Case. Discrete Form -- 74.4.6 Summary and List of Further Reading -- 74.5 Maximum Entropy and Related Probabilistic Methods -- 74.6 Exit-Wave Reconstruction -- 75 Three-Dimensional Reconstruction -- 75.1 Introduction -- 75.2 Methods -- 75.2.1 Direct Methods -- 75.2.1.1 Discrete Data -- 75.2.2 Iterative Methods -- 75.2.3 Reconstruction From a Single View of an Oblique Section -- 75.2.4 Ptycho-Tomography -- 75.2.5 The Missing Wedge or Cone -- 75.2.6 Compressed Sensing -- 75.2.6.1 Introduction -- 75.2.6.2 Sparsity -- 75.2.6.3 Applied Compressed Sensing -- 75.2.6.4 Electron Tomography -- 75.2.7 Breakdown of the Projection Requirement -- Artificial Neural Networks -- 75.2.8 Reconstruction Quality -- 75.3 Preprocessing -- 75.3.1 Background -- 75.3.2 Alignment -- 75.3.3 Classification by Correspondence Analysis -- 75.3.4 Random Tilt Series -- 75.3.5 Removal of Distortion -- 75.3.6 Defocus Gradient
• 70.2.4 Operations Involving Images and Templates -- 70.2.4.1 General Definition -- 70.2.4.2 Special Cases -- 70.2.5 Concluding Remarks -- 70.3 Notation -- 71 Acquisition, Sampling and Coding -- 71.1 Acquisition -- 71.2 Sampling -- 71.2.1 The Sampling Theorem -- 71.2.2 Degrees of Freedom -- 71.3 Quantization -- 71.4 Coding -- 71.4.1 Use of Image Transforms -- 71.4.2 Predictive Coding -- 71.4.3 Huffman and Vector Codes -- 71.5 Electron Optical Considerations -- 71.5.1 Acquisition -- 71.5.2 Sampling -- 72 Enhancement -- 72.1 Operations on Individual Pixels -- 72.1.1 Elementary Operations -- 72.1.2 Histogram-Based Enhancement -- 72.1.2.1 Practical Difficulties -- 72.1.2.2 Refinements -- 72.2 Linear Filtering -- 72.2.1 Low-Pass Filters -- 72.2.2 High-Pass Filters -- 72.2.3 Hexagonal Sampling -- 72.2.4 Generalized Convolution -- 72.2.5 Periodic Specimens -- 72.3 Nonlinear Filters -- 72.3.1 Nonlinear Exploitation of Linear Filtering -- 72.3.2 Median and Rank-Order Filtering -- 72.3.3 Morphological Filters -- 72.3.3.1 Introduction -- 72.3.3.2 Binary Image Morphology -- 72.3.3.3 Interpretation in Terms of Convolution -- 72.3.3.4 Combinations of Dilation and Erosion -- 72.3.3.5 Grey-Level Image Morphology -- 72.3.3.6 Practical Applications -- 72.3.3.7 Linearity -- 72.4 Image Algebraic Representation of Enhancement -- 72.4.1 Formation of the Histogram -- 72.4.2 Convolutional Filters -- 72.5 Enhancement in Electron Microscopy -- 73 Linear Restoration -- 73.1 Introduction -- 73.2 Extended Wiener Filters -- 73.3 Filtering With Constraints -- 73.4 Hoenders' Procedure -- 73.5 Recursive Filtering -- 73.6 Other Approaches -- 74 Nonlinear Restoration - The Phase Problem -- 74.1 Introduction -- 74.1.1 Formal Statement of the Problem -- 74.2 Extended Linear Approximation -- 74.3 Multiple Recordings (Circular Symmetry) -- 74.3.1 Introduction