Principles of Electron Optics

Transcription

1 Principles of Electron Optics Volume 1 Basic Geometrical Optics by P. W. HAWKES CNRS Laboratory of Electron Optics, Toulouse, France and E. KASPER Institut für Angewandte Physik Universität Tübingen, Federal Republic of Germany 1989 ACADEMIC PRESS Harcourt Brace lovanovich, Publishers London San Diego New York Berkeley Boston Sydney Tokyo Toronto

2 Contents of Volume 1 Basic Geometrical Optics Preface 1 Introduction Organization of the subject History 8 PART I - CLASSICAL MECHANICS 2 Relativistic Kinematics The Lorentz equation and general considerations Conservation of energy The acceleration potential Definition of coordinate systems Conservation of axial angular momentum 24 3 Different Forms of Trajectory Equations Parametric representation in terms of the arc-length Relativistic proper-time representation The cartesian representation Scaling rules 33 4 Variational Principles The Lagrange formalism General rotationally symmetric systems The canonical formalism The time-independent form of the variational principle Static rotational ly symmetric systems 44 5 Hamiltonian Optics Introduction of the characteristic function The Hamilton-Jacobi equation The analogy with light optics The influence of vector potentials Gauge transformations 53

3 VI Contents 5.6 Poincare's integral invariant The problem of uniqueness Resume 58 PART II - CALCULATION OF STATIC FIELDS 6 Basic Concepts and Equations General considerations Field equations Variational principles Rotationally symmetric fields Planar fields 69 7 Series Expansions Azimuthal Fourier series expansions Radial series expansions Rotationally symmetric fields Multipole fields Planar fields Fourier-Bessel series expansions 91 8 Boundary-Value Problems Boundary-value problems in electrostatics Boundary conditions in magnetostatics Examples of boundary-value problems in magnetostatics Integral Equations Integral equations for scalar potentials Problems with interface conditions Reduction of the dimensions Important special cases Resume The Boundary-Element Method Evaluation of the Fourier integral kernels Numerical solution of one-dimensional integral equations Superposition of aperture fields Three-dimensional Dirichlet problems Examples of applications of the boundary-element method The Finite-Difference Method (FDM) The choice of grid 159

4 Contents VII 11.2 The Taylor series method The integration method Nine-point formulae Iterative solution techniques The Finite-Element Method (FEM) Formulation for round magnetic lenses Formulation for self-adjoint elliptic equations Solution of the finite-element equations Improvementof the finite-element method Comparison and combination of different methods Field-Interpolation Techniques One-dimensional differentiation and interpolation Two-dimensional interpolation 194 PART III-THE PARAXIAL APPROXIMATION 14 Introduction Systems with an Axis of Rotational Symmetry Derivation of the paraxial ray equations from the general ray equations Variational derivation of the paraxial equations Forms of the paraxial equations and general properties of their solutions The Abbe sine condition and Herschel's condition Some other transformations Gaussian Optics of Rotationally Symmetric Systems: Asymptotic Image Formation Real and asymptotic image formation Asymptotic cardinal elements and transfer matrices Gaussian optics as a projective transformation Use of the angle characteristic to establish the optical quantities 237

5 VIII Contents 16.5 The existence of asymptotes Gaussian Optics of Rotationally Symmetric Systems: Real Cardinal Elements Real cardinal elements for high magnification and high demagnification Osculating cardinal elements Inversion of the principal planes Approximate formulae for the cardinal elements: the thin-lens approximation and the weak-lens approximation Electron Mirrors Introduction A time-like parameter as independent variable The cartesian representation A quadratic transformation Quadrupole Lenses Paraxial equations for quadrupoles Transaxial lenses Cylindrical Lenses 290 PART IV - ABERRATIONS Introduction Perturbation Theory: General Formalism The Relation Between Permitted Types of Aberration and System Symmetry Introduction N= 1 N = 2 N = 3 N = 4 N = 5 and 6 Systems with an axis of rotational symmetry Note on the classification of aberrations The Geometrical Aberrations of Round Lenses Introduction

6 Contents ix 24.2 Derivation of the real aberration coefficients Spherical aberration Coma Astigmatism and field curvature Distortion The variation of the aberration coefficients with aperture position Reduced coordinates Seman's transformation of the characteristic function Asymptotic Aberration Coefficients Chromatic Aberrations Real chromatic aberrations Asymptotic chromatic aberrations Aberration Matrices and the Aberrations of Lens Combinations The Aberrations of Mirrors and Cathode Lenses The parametric form of the theory Systems with curved cathodes Structure of the aberrations The cartesian form of the aberration theory The Aberrations of Quadrupole Lenses and Octopoles Introduction Geometrical aberration coefficients Aperture aberrations Chromatic aberrations Quadrupole multiplets The Aberrations of Cylindrical Lenses Parasitic Aberrations Small deviations from rotational symmetry; axial astigmatism Classification of the parasitic aberrations Numerical determination of parasitic aberrations The isoplanatic approximation 477

7 X Contents PART V- DEFLECTION SYSTEMS 32 Deflection Systems and their Aberrations Introduction The paraxial optics of deflection systems The aberrations of deflection systems Stigmators 516 PART VI - COMPUTER-AIDED ELECTRON OPTICS 33 Numerical Calculation of Trajectories, Paraxial Properties and Aberrations Introduction Numerical solution of ordinary differential equations Standard applications in electron optics Differential equations for the aberrations Least-squares-fit methods in electron optics Determination and evaluation of aberration discs Optimization procedures The Use of Computer Algebra Languages Introduction Computer algebra, its role in electron optics Two practical examples 569 Notes and References Preface and Part I, s Part II, s Part III, s Part IV, s Part V, Part VI, s 33 and Conference Proceedings 1192 Index