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
Contents of Volume 1 Basic Geometrical Optics Preface 1 Introduction 1 1.1 Organization of the subject 3 1.2 History 8 PART I - CLASSICAL MECHANICS 2 Relativistic Kinematics 17 2.1 The Lorentz equation and general considerations 17 2.2 Conservation of energy 18 2.3 The acceleration potential 19 2.4 Definition of coordinate systems 22 2.5 Conservation of axial angular momentum 24 3 Different Forms of Trajectory Equations 27 3.1 Parametric representation in terms of the arc-length 27 3.2 Relativistic proper-time representation 29 3.3 The cartesian representation 30 3.4 Scaling rules 33 4 Variational Principles 35 4.1 The Lagrange formalism 35 4.2 General rotationally symmetric systems 38 4.3 The canonical formalism 41 4.4 The time-independent form of the variational principle 43 4.5 Static rotational ly symmetric systems 44 5 Hamiltonian Optics 46 5.1 Introduction of the characteristic function 46 5.2 The Hamilton-Jacobi equation 48 5.3 The analogy with light optics 49 5.4 The influence of vector potentials 51 5.5 Gauge transformations 53
VI Contents 5.6 Poincare's integral invariant 54 5.7 The problem of uniqueness 57 5.8 Resume 58 PART II - CALCULATION OF STATIC FIELDS 6 Basic Concepts and Equations 61 6.1 General considerations 61 6.2 Field equations 62 6.3 Variational principles 65 6.4 Rotationally symmetric fields 67 6.5 Planar fields 69 7 Series Expansions 73 7.1 Azimuthal Fourier series expansions 73 7.2 Radial series expansions 78 7.3 Rotationally symmetric fields 85 7.4 Multipole fields 88 7.5 Planar fields 90 7.6 Fourier-Bessel series expansions 91 8 Boundary-Value Problems 94 8.1 Boundary-value problems in electrostatics 94 8.2 Boundary conditions in magnetostatics 96 8.3 Examples of boundary-value problems in magnetostatics 101 9 Integral Equations 107 9.1 Integral equations for scalar potentials 107 9.2 Problems with interface conditions 111 9.3 Reduction of the dimensions 113 9.4 Important special cases 117 9.5 Resume 124 10 The Boundary-Element Method 125 10.1 Evaluation of the Fourier integral kernels 125 10.2 Numerical solution of one-dimensional integral equations 131 10.3 Superposition of aperture fields 143 10.4 Three-dimensional Dirichlet problems 149 10.5 Examples of applications of the boundary-element method 158 11 The Finite-Difference Method (FDM) 159 11.1 The choice of grid 159
Contents VII 11.2 The Taylor series method 160 11.3 The integration method 162 11.4 Nine-point formulae 165 11.5 Iterative solution techniques 170 12 The Finite-Element Method (FEM) 175 12.1 Formulation for round magnetic lenses 1 75 12.2 Formulation for self-adjoint elliptic equations 179 12.3 Solution of the finite-element equations 182 12.4 Improvementof the finite-element method 183 12.5 Comparison and combination of different methods 184 13 Field-Interpolation Techniques 188 13.1 One-dimensional differentiation and interpolation 188 13.2 Two-dimensional interpolation 194 PART III-THE PARAXIAL APPROXIMATION 14 Introduction 201 15 Systems with an Axis of Rotational Symmetry 202 15.1 Derivation of the paraxial ray equations from the general ray equations 207 15.2 Variational derivation of the paraxial equations 212 15.3 Forms of the paraxial equations and general properties of their solutions 213 15.4 The Abbe sine condition and Herschel's condition 219 15.5 Some other transformations 222 16 Gaussian Optics of Rotationally Symmetric Systems: Asymptotic Image Formation 225 16.1 Real and asymptotic image formation 225 16.2 Asymptotic cardinal elements and transfer matrices 226 16.3 Gaussian optics as a projective transformation 235 16.4 Use of the angle characteristic to establish the optical quantities 237
VIII Contents 16.5 The existence of asymptotes 239 17 Gaussian Optics of Rotationally Symmetric Systems: Real Cardinal Elements 242 17.1 Real cardinal elements for high magnification and high demagnification 242 17.2 Osculating cardinal elements 246 17.3 Inversion of the principal planes 253 17.4 Approximate formulae for the cardinal elements: the thin-lens approximation and the weak-lens approximation 257 18 Electron Mirrors 261 18.1 Introduction 261 18.2 A time-like parameter as independent variable 264 18.3 The cartesian representation 271 18.4 A quadratic transformation 274 19 Quadrupole Lenses 276 19.1 Paraxial equations for quadrupoles 277 19.2 Transaxial lenses 286 20 Cylindrical Lenses 290 PART IV - ABERRATIONS 21 22 23 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 24 24.1 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 297 303 315 315 322 325 329 330 333 334 336 339 339
Contents ix 24.2 Derivation of the real aberration coefficients 339 24.3 Spherical aberration 350 24.4 Coma 365 24.5 Astigmatism and field curvature 369 24.6 Distortion 378 24.7 The variation of the aberration coefficients with aperture position 382 24.8 Reduced coordinates 384 24.9 Seman's transformation of the characteristic function 387 25 Asymptotic Aberration Coefficients 393 26 Chromatic Aberrations 409 26.1 Real chromatic aberrations 409 26.2 Asymptotic chromatic aberrations 415 27 Aberration Matrices and the Aberrations of Lens Combinations 418 28 The Aberrations of Mirrors and Cathode Lenses 425 28.1 The parametric form of the theory 425 28.2 Systems with curved cathodes 429 28.3 Structure of the aberrations 430 28.4 The cartesian form of the aberration theory 432 29 The Aberrations of Quadrupole Lenses and Octopoles 434 29.1 Introduction 434 29.2 Geometrical aberration coefficients 434 29.3 Aperture aberrations 453 29.4 Chromatic aberrations 460 29.5 Quadrupole multiplets 461 30 The Aberrations of Cylindrical Lenses 466 31 Parasitic Aberrations 470 31.1 Small deviations from rotational symmetry; axial astigmatism 470 31.2 Classification of the parasitic aberrations 472 31.3 Numerical determination of parasitic aberrations 475 31.4 The isoplanatic approximation 477
X Contents PART V- DEFLECTION SYSTEMS 32 Deflection Systems and their Aberrations 483 32.1 Introduction 483 32.2 The paraxial optics of deflection systems 487 32.3 The aberrations of deflection systems 497 32.4 Stigmators 516 PART VI - COMPUTER-AIDED ELECTRON OPTICS 33 Numerical Calculation of Trajectories, Paraxial Properties and Aberrations 525 33.1 Introduction 525 33.2 Numerical solution of ordinary differential equations 526 33.3 Standard applications in electron optics 533 33.4 Differential equations for the aberrations 537 33.5 Least-squares-fit methods in electron optics 543 33.6 Determination and evaluation of aberration discs 547 33.7 Optimization procedures 558 34 The Use of Computer Algebra Languages 565 34.1 Introduction 565 34.2 Computer algebra, its role in electron optics 566 34.3 Two practical examples 569 Notes and References Preface and 1 575 Part I, s 2-5 586 Part II, s 6-13 586 Part III, s 14-20 591 Part IV, s 21-31 601 Part V, 32 619 Part VI, s 33 and 34 622 Conference Proceedings 1192 Index