Microdevices

1. Introductory Survey.- 1-1. Introduction.- 1-2. Electromagnetic Fields.- 1-2-1. Maxwell's Equations.- 1-2-2. Static Fields.- 1-2-3. Stokes's Theorem.- 1-3. Some Basic Classical Mechanics.- 1-3-1. Hamilton's Principle; The Lagrangian Equations of Motion.- 1-3-2. The Maupertuis Principle.- 1-4. A Little Reminder of Geometrical Optics.- 1-4-1. Fermat's Principle; The Index of Refraction.- 1-4-2. Axially Symmetric Lenses.- Summary.- 2. Motion of Charged Particles in Electric and Magnetic Fields.- 2-1. The Lagrangian.- 2-2. Conservation of Energy.- 2-2-1. Motion of Free Particles; Velocity versus Potential.- 2-3. The Equations of Motion.- 2-4. The Trajectory Equations.- 2-5. The Relativistic Potential.- 2-6. The Electron Optical Index of Refraction.- 2-7. Particles in Homogeneous Fields.- 2-7-1. The Parallel-Plate Capacitor.- 2-7-1-1. Electrostatic Deflection.- 2-7-1-2. A Simple Velocity Analyzer.- 2-7-2. Homogeneous Magnetic Field.- 2-7-2-1. Long Magnetic Lens.- 2-7-2-2. Magnetic Deflection.- 2-7-3. The Simultaneous Action of Homogeneous Electric and Magnetic Fields.- 2-7-3-1. Mass Analysis and Other Applications.- 2-8. Scaling Laws.- Summary.- 3. Determination of Electric and Magnetic Fields.- 3-1. Analytical Methods.- 3-1-1. Series Expansions of Potentials and Fields.- 3-1-1-1. Planar Fields.- 3-1-1-2. Axially Symmetric Fields.- 3-1-1-3. Multipole Fields.- 3-1-2. Analytical Calculation of Axially Symmetric Potential Fields.- 3-1-2-1. Separation of Variables.- 3-1-2-2. Difficulties of Analytical Calculations (Electrostatic Field of Two Equidiameter Cylinders).- 3-1-2-3. Field of a Circular Aperture.- 3-1-2-4. Rapid Evaluation of Fields Produced by Two or More Circular Apertures.- 3-1-3. Analytical Calculation of Multipole Fields.- 3-1-3-1. Short Multipoles.- 3-1-3-2. Long Multipoles.- 3-1-3-3. Ideal Multipoles.- 3-1-3-4. The Method of Conformal Transformation.- 3-1-4. On the Role of Magnetic Materials.- 3-1-5. Analytical Calculation of Magnetic Fields Produced by Currents.- 3-1-5-1. The Biot-Savart Law.- 3-1-5-2. Field of a Straight Wire.- 3-1-5-3. Field of a Circular Loop.- 3.1.5.4. Field of a Thin Solenoid.- 3-1-5-5. Field of a Multilayer Coil.- 3-1-5-6. Field of a Pancake Coil.- 3-2. Measurement of Fields and Analog Methods.- 3-2-1. Measurement of Magnetic Fields.- 3-2-1-1. Electromagnetic Induction.- 3-2-1-2. Hall Effect.- 3-2-1-3. Permalloy and Bismuth Probes.- 3-2-1-4. Magnetic Resonance.- 3-2-2. Analog Methods.- 3-2-2-1. The Electrolytic Tank.- 3-2-2-2. The Resistor Network.- 3-2-2-3. Other Analog Methods.- 3-3. Numerical Methods.- 3-3-1. Accuracy.- 3-3-1-1. Errors Due to the Nature of the Problem.- 3-3-1-2. Errors Due to the Number Representation in the Computer.- 3-3-1-3. Errors Due to the Numerical Method.- 3-3-2. The Finite-Difference Method.- 3-3-2-1. Methods of Solution for Systems of Algebraic Equations.- 3-3-3. The Finite-Element Method.- 3-3-4. The Charge-Density (Integral) Method.- 3-3-5. Numerical Differentiation and Interpolation.- 3-3-5-1. Differentiation.- 3-3-5-2. Lagrange Interpolation.- 3-3-5-3. The Interpolating Pulse.- 3.3.5.4. The Cubic Spline.- Summary.- 4. Focusing With Axially Symmetric Fields.- 4-1. Busch's Theorem.- 4-2. The General Trajectory Equation.- 4-3. The Paraxial Ray Equation.- 4-4. Image Formation by Paraxial Rays.- 4-5. The Helmholtz-Lagrange Formula.- 4-6. Cardinal Elements.- 4-6-1. Asymptotic Cardinal Elements.- 4-7. Electron and Ion Lenses.- 4-8. Systems of Lenses.- 4-8-1. The Transfer Matrix.- 4-8-2. Combination of Two Thick Lenses.- 4-9. The Thin-Lens Approximation.- 4-9-1. Combination of Thin Lenses.- 4-10. Examples of Paraxial Focusing.- 4-10-1. Paraxial Trajectories in Homogeneous Fields.- 4-10-1-1. Homogeneous Electrostatic Field.- 4-10-1-2. Skew Rays.- 4-10-1-3. Homogeneous Magnetic Field.- 4-10-2. The Single-Loop Magnetic Lens.- 4-10-3. Lens Systems.- 4-10-3-1. Telescopic System.- 4-10-3-2. Magnification of Lens Systems.- Summary.- 5. The Theory of Aberrations.- 5-1. The Method of Characteristic Functions.- 5-2. Geometrical Aberrations.- 5-2-1. Spherical Aberration.- 5-2-1-1. Zero and Infinite Magnifications.- 5-2-1-2. Alternative Forms of the Spherical Aberration Coefficient.- 5-2-1-3. Scherzer's Theorem.- 5-2-1-4. The Disk of Minimum Confusion.- 5-2-2. Astigmatism.- 5-2-3. Curvature of Field.- 5-2-4. Distortion.- 5-2-5. Coma.- 5-2-6. Anisotropic Aberrations.- 5-2-6-1. Anisotropic Astigmatism.- 5-2-6-2. Anisotropic Distortion.- 5-2-6-3. Anisotropic Coma.- 5-2-7. On the Relative Importance of the Different Geometrical Aberrations.- 5-3. Chromatic Aberration.- 5-3-1. Axial Chromatic Aberration.- 5-3-1-1. Zero and Infinite Magnifications.- 5-3-1-2. The Upper Limit of the Axial Chromatic Aberration.- 5-3-2. Chromatic Aberration of Magnification.- 5-3-3. Anisotropic Chromatic Aberration.- 5-3-4. Magnetic Chromatic Aberration.- 5-4. Asymptotic Aberrations.- 5-4-1. The Dependence of the Asymptotic Aberration Coefficients on the Magnification.- 5-4-1-1. Polynomial Expression for the Asymptotic Spherical Aberration Coefficient.- 5-4-1-2. Polynomial Expression for the Asymptotic Axial Chromatic Aberration Coefficient.- 5-4-2. Aberrations of Thin Lenses.- 5-4-2-1. Spherical Aberration.- 5-4-2-2. Axial Chromatic Aberration.- 5-5. Aberrations of Lens Combinations.- 5-5-1. Addition of Spherical Aberrations.- 5-5-2. Addition of Axial Chromatic Aberrations.- 5-6. Other Sources of Aberrations and Aberration Correction.- 5-6-1. Diffraction.- 5-6-2. Space Charge and Surface Charges.- 5-6-3. High-Frequency Fields.- 5-6-4. Lack of Axial Symmetry.- 5-6-5. Other Methods of Correction.- 5-6-5-1. Coaxial Lenses.- 5-6-5-2. Symmetric Trajectories.- 5-6-5-3. Position of the Limiting Aperture.- 5-6-5-4. Digital Image Processing.- 5-6-6. Synthesis.- 5-6-7. On the Measurement of Aberrations.- 5-6-8. Brightness.- 5-7. Simultaneous Action of Different Aberrations.- 5-7-1. Negligibly Small Sources.- 5-7-2. Finite Sources.- 5-7-2-1. Negligible Chromatic Aberration.- 5-7-2-2. Negligible Spherical Aberration.- 5-7-3. Aberration Mixing for Lens Combinations.- 5-7-4. Figures of Merit.- Summary.- 6. Numerical Techniques for Ray Tracing and Calculation of Aberrations.- 6-1. Analytical Models.- 6-2. Numerical Ray Tracing.- 6-2-1. The Runge-Kutta Method.- 6-2-2. Multistep Methods.- 6-2-2-1. Numerov's Method.- 6-2-3. Additional Remarks on Accuracy.- 6-3. Numerical Calculation of Aberration Integrals.- 6-3-1. Trapezoidal Integration.- 6-3-2. Simpson's Rule.- 6-3-3. Romberg Integration and the Gaussian Quadrature.- Summary.- 7. Electrostatic Lenses.- 7-1. General Properties and Relationships.- 7-2. Electrostatic Lens Models.- 7-2-1. Analytical Models.- 7-2-2. The Piecewise Linear Model.- 7-2-3. The Piecewise Quadratic Model.- 7-2-4. The Spline Model.- 7-3. Two-Electrode Immersion Lenses.- 7-3-1. Geometrically Symmetric Lenses.- 7-3-1-1. A Linear Model.- 7-3-1-2. An Analytical Model.- 7-3-1-3. The Two-Cylinder Lens.- 7-3-1-4. The Double-Aperture Lens.- 7-3-1-5. Polynomial Lenses.- 7-3-2. Asymmetric Lenses.- 7-3-2-1. Analytical Models.- 7-3-2-2. The Asymmetric Two-Cylinder Lens.- 7-3-2-3. A Hybrid Lens.- 7-4. Unipotential Lenses.- 7-4-1. Symmetric Lenses.- 7-4-1-1. A Piecewise Linear Model.- 7-4-1-2. A Piecewise Quadratic Model.- 7-4-1-3. An Analytical Model.- 7-4-1-4. The Three-Cylinder Lens.- 7-4-1-5. The Triple-Aperture Lens.- 7-4-1-6. Other Types of Symmetric Lenses.- 7-4-2. Asymmetric Lenses.- 7-5. Three-Electrode Immersion Lenses.- 7-5-1. Geometrically Symmetric Lenses.- 7-5-1-1. The Three-Cylinder Lens.- 7-5-1-2. Other Types of Geometrically Symmetric Lenses.- 7-5-2. Asymmetric Lenses.- 7-6. Multielectrode Lenses.- 7-6-1. Four-Electrode Lenses.- 7-6-2. Lenses with Five or More Electrodes.- 7-6-3. Spline Lenses.- 7-7. Comparison of Different Electrostatic Lenses.- 7-8. Lenses Immersed in Fields.- 7-8-1. The Exponential Model.- 7-8-2. The Single-Aperture Lens.- 7-8-3. Cathode Lenses, Electron and Ion Sources.- 7-8-3-1. Thermionic Guns.- 7-8-3-2. Field-Emission Guns.- 7-8-3-3. Ion Sources.- Summary.- 8. Magnetic Lenses.- 8-1. General Properties and Relationships.- 8-2. Long Lenses.- 8-2-1. Homogeneous Magnetic Fields.- 8-2-2. Linear Magnetic Fields.- 8-2-3. Long Lenses with Low Spherical Aberration.- 8-3. Magnetic Lens Models.- 8-3-1. The Rectangular Model.- 8-3-2. The Step-Function Model.- 8-3-3. The Piecewise Linear Model.- 8-3-4. The Spline Model.- 8-3-5. Glaser's Bell-Shaped Model.- 8-3-5-1. Generalization of the Bell-Shaped Model.- 8-3-6. The Grivet-Lenz Model.- 8-3-7. Other Models.- 8-4. Short Lenses.- 8-4-1. Conventional Lenses.- 8-4-2. Unconventional Lenses.- 8-4-2-1. Superconducting Lenses.- 8-4-2-2. Reduction of the Coil Size by Other Means.- 8-4-2-3. Rotation-Free Miniature Lenses.- 8-4-2-4. Iron-Free Magnetic Lenses.- 8-4-2-5. Single Pole-Piece Lenses.- Summary.- 9. Computer-Aided Optimization and Synthesis of Electron and Ion Lenses.- 9-1. Is Aberrationless Electron/Ion Optics Possible?.- 9-1-1. The Lower Limit of the Axial Chromatic Aberration of Magnetic Lenses.- 9-2. Optimization: Synthesis versus Analysis.- 9-3. Early Attempts of Synthesis.- 9-4. Calculus of Variations.- 9-4-1. The Lower Limits of the Spherical and Axial Chromatic Aberration Coefficients.- 9-5. Dynamic Programming.- 9-6. Optimal Control Procedure.- 9-7. Analytical Functions.- 9-8. Reconstruction of Electrodes and Pole Pieces from the Optimized Axial Field Distributions.- 9-9. Polynomial and Spline Lenses.- 9-9-1. Polynomial Lenses.- 9-9-2. Spline Lenses.- 9-9-2-1. Two-Interval Spline Lenses.- 9-10. The Synthesis Procedure.- 9-10-1. Application: Unconventional Electrostatic Lenses.- 9-11. Artificial Intelligence Techniques.- Summary.- 10. Multipole Lenses.- 10-1. The Fields of Multipole Lenses.- 10-2. The Paraxial Ray Equations.- 10-3. Image Formation by Paraxial Rays.- 10-4. Systems of Quadrupoles.- 10-4-1. Transfer Matrices.- 10-4-2. Thin-Lens Representation.- 10-4-3. Doublets.- 10-4-4. Triplets.- 10-4-5. Multiplets.- 10-4-5-1. Beam Matching.- 10-5. Aberrations of Multipole Lenses.- 10-5-1. Geometrical Aberrations.- 10-5-2. Correction of Aberrations by Means of Multipoles.- 10-5-3. Chromatic Aberration.- 10-5-3-1. The Achromatic Quadrupole Lens.- Summary.- 11. Beam Deflection.- 11-1. Deflection for Scanning.- 11-1-1. Electrostatic Deflection Fields.- 11-1-2. Magnetic Deflection Fields.- 11-1-3. Stigmatic Imaging with Small Deflection.- 11-1-4. Deflection Aberrations.- 11-2. Electrostatic and Magnetic Prisms.- 11-2-1. Electrostatic Prisms.- 11-2-2. Magnetic Prisms.- 11-3. New Symmetries-New Possibilities.- Summary.- 12. High-Intensity Beams.- 12-1. Space-Charge Optics.- 12-1-1. Space-Charge Forces.- 12-1-1-1. The Electrostatic Force.- 12-1-1-2. The Magnetic Force.- 12-1-2. Beam Spreading.- 12-1-3. Production of High-Intensity Beams.- 12-1-3-1. Space-Charge Flow.- 12-1-3-2. The Pierce Gun.- 12-1-4. Maintenance of High-Intensity Beams.- 12-1-4-1. Focusing by Homogeneous Magnetic Fields.- 12-1-4-2. Periodic Focusing.- 12-2. The Boersch Effect.- Summary.- References.