Electron and Ion Optics

The field of electron and ion optics is based on the analogy between geometrical light optics and the motion of charged particles in electromagnetic fields. The spectacular development of the electron microscope clearly shows the possibilities of image formation by charged particles of wavelength much shorter than that of visible light. As new applications such as particle accelerators, cathode ray tubes, mass and energy spectrometers, microwave tubes, scanning-type analytical instruments, heavy beam technologies, etc. emerged, the scope of particle beam optics has been exten­ ded to the formation of fine probes. The goal is to concentrate as many particles as possible in as small a volume as possible. Fabrication of microcircuits is a good example of the growing importance of this field. The current trend is towards increased circuit complexity and pattern density. Because of the diffraction limitation of processes using optical photons and the technological difficulties connected with x-ray processes, charged particle beams are becoming popular. With them it is possible to write directly on a wafer under computer control, without using a mask. Focused ion beams offer especially great possibilities in the submicron region. Therefore, electron and ion beam technologies will most probably playa very important role in the next twenty years or so.

Contenu

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 …