Electron Beams, Lenses, and Optics

Electron Beams, Lenses, and Optics, Volume I deals with the physics of electron beams, lenses, and optics and covers topics ranging from the paraxial ray in symmetrical electric fields to the analytical determination of electrostatic fields. The general properties of electrostatic lenses and the electrostatic immersion lens are also considered. Each equation except one is derived from first principles. To emphasize the physics of the discussions, elementary mathematics is used as much as possible.

Comprised of eight chapters, this volume begins with an introduction to the laws that govern electron beams and light rays, including Snell's law. Some fundamental limitations to the analogy between electron optics and light optics are evaluated, together with electron rays in plane symmetrical and in rotationally symmetrical fields. Subsequent chapters explore the general properties of electrostatic lenses and electrostatic immersion lenses; electrostatic unipotential lenses; and formation of optical images by rotationally symmetrical magnetic fields. The final chapter is devoted to the symmetrical magnetic lens and its magnetic scalar potential, paying particular attention to the flux density along the z axis and factors to consider in the design of the pole pieces.

This book will be of interest to students, practitioners, and researchers in physics.

Contenu

Preface

List of Symbols

I. Electron Beams and Light Rays

1.1 Snell's Law

1.2 General Properties of the Two Optical Media

1.3 Fermat's Principle

1.4 Hamilton's Principle

1.5 Limitations of the Analogy

Further Reading

II. The Paraxial Ray in Symmetrical Electric Fields

2.1 Plane Symmetrical Fields

2.2 Paraxial Rays in Plane Symmetrical Electric Fields

2.3 Rotationally Symmetrical Systems

2.4 Paraxial Rays in Rotationally Symmetrical Electric Fields

2.5 Picht's Reduced Formula

2.6 Image Formation by Electric Fields

2.7 Image Formation by Paraxial Rays in Rotationally Symmetrical Fields with Superimposed Deflection

2.8 The Radius of Curvature as a Function of the Potential

2.9 Determination of the Second Derivative of the Potential

III. Analytical Determination of Electrostatic Fields

3.1 Laplace's Equation

3.2 Potential Distribution by Means of Complex Functions

3.3 Method of Separation of Variables

3.4 Conformal Transformation

3.5 Axial Potential of Two Equidiameter Cylinders with Negligible Separation

3.6 Axial Potential of Two Coaxial Equidiameter Cylinders Separated by a Distance

3.7 An Empirical Relation

3.8 Potential Distribution Due to a Circular Hole Separating Two Uniform Fields

3.9 Condition at the Aperture

3.10 Condition at the Saddle Point

Further Reading

IV. General Properties of Electrostatic Lenses

4.1 Cardinal Points of a Lens

4.2 Important Lens Relations

4.3 Newton's Formula

4.4 Sign Convention

4.5 Weak and Strong Lenses

4.6 The Law of Helmholtz-Lagrange

4.7 Lens Relations in Electron Optics

4.8 The Action of an Electrostatic Lens

4.9 The Types of Electrostatic Lenses

4.10 Focal Length of the Weak Electrostatic Lens

4.11 Focal Length Starting with the Reduced Formula

4.12 Position of Principal Planes

4.13 Combination of Thin Lenses

4.14 The Cardinal Points of a Strong Lens

4.15 Method of Sectionizing the Lens

4.16 Method of Successive Approximation

4.17 Development up to Second Approximation

V. The Electrostatic Immersion Lens

5.1 Symmetrical Two-Cylinder Lens with Negligible Gap

5.2 Method of Sectionizing the Lens

5.3 Analysis of the Symmetrical Two-Cylinder Lens Using a Digital Computer

5.4 Analysis of the Results

5.5 The Asymmetrical Two-Tube Lens

5.6 Analysis of the Results

Appendix A5

Further Reading

VI. The Electrostatic Unipotential Lens

6.1 Axial Potential Distribution

6.2 Potentials at the Centers of the Inner and Outer Electrodes

6.3 Potential Configuration along the Axis

6.4 Ray Equations in the Regions of Interest

6.5 Evaluation of the Constants

6.6 Focal Length

6.7 Midfocal Length

6.8 Principal Plane

6.9 Trajectories in the Three-Aperture Lens

6.10 Course of the Trajectories in the Regions of Interest of the Three-Aperture Lens

6.11 Design Examples

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6.13 Experimental Results

Further Reading

VII. Optical Image Formation by Rotationally Symmetrical Magnetic Fields

7.1 The Homogeneous Magnetic Field

7.2 The Inhomogeneous Magnetic Field

7.3 Series Expansion of the Magnetic Field

7.4 Paraxial-Ray Equation

7.5 Image Rotation

7.6 Busch's Equation for a Weak Lens

7.7 The Glaser Model

7.8 The Focal Points

7.9 The Focal Lengths

7.10 The Principal Planes

7.11 Object and Image Relationship

7.12 Magnification

7.13 Image Rotation

7.14 The Parameter K2

7.15 Comparison of Glaser's Result with Busch's Equation

7.16 The Magnetic Lens with Unsymmetrical Pole Pieces

7.17 Accuracy of the Glaser Model

7.18 Numerical Example Using Glaser's Formulas

7.19 Combined Electric and Magnetic Fields in Rotationally Symmetrical Systems

7.20 Theorem of Larmor

7.21 Special Case: is Constant

Further Reading

VIII. The Symmetrical Magnetic Lens

8.1 The Short Air Coil

8.2 Focal Length of an Air Coil

8.3 The Shielded Magnetic Lens

8.4 Magnetic Scalar Potential of a Symmetrical Lens

8.5 The Flux Density along the z Axis

8.6 Considerations in the Design of the Pole Pieces

8.7 Analysis of the Symmetrical Magnetic Lens

Further Reading