Contents
Preface
ix

1 Introduction: The optical nature of a charged particle beam
1

2 Geometrical optics
19

2.1 Relativistic classical mechanics . . . . . . . . . . . . 20

2.1.1 Hamilton’s principle of least action . . . . . 21

2.1.2 The Hamiltonian function and energy conservation . . . . . . . . . . . . . . . . . . . . 26

2.1.3 Mechanical analog of Fermat’s principle . . . 28

2.2 Exact trajectory equation for a single particle . . . 32

2.3 Conservation laws . . . . . . . . . . . . . . . . . . . 34

2.3.1 The Lagrange invariant . . . . . . . . . . . . 35

2.3.2 Liouville’s theorem and brightness conservation . . . . . . . . . . . . . . . . . . . . . . 41

2.4 General curvilinear axis . . . . . . . . . . . . . . . 45

2.4.1 Equation of motion in terms of transverse

coordinates and slopes . . . . . . . . . . . . 46

2.4.2 Natural units . . . . . . . . . . . . . . . . . 49

2.5 Axial symmetry . . . . . . . . . . . . . . . . . . . . 51

2.5.1 Exact equations of motion for axially symmetric fields . . . . . . . . . . . . . . . . . . 51

2.5.2 Paraxial approximation, Gaussian optics . . 54

2.5.3 Series solution for the general ray equation . 57

2.5.4 Space charge . . . . . . . . . . . . . . . . . 63

2.5.5 The primary geometrical aberrations . . . . 68

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