First - and Second - Order Charged Particle Optics

2 A General Firstand Second-Order Theory of Beam Transport Optics . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 The Vector Differential Equation of Motion . . . . . . . . 2.2 The Coordinate System . . . . . . . . . . . . . . . . .’ . 2.3 Expanded Form of a Magnetic Field Having Median Plane Symmetry . . . . . . . . . . . . . . . . . . . . . . 2.4 Field Expansion to Second Order Only . . . . . . . . . . 2.5 Expanded of the Magnetic Field as a Function of Multipole Components . . . . . . . . . . . . . . . . . . . 2.5.1 Multipole Strengths for Pure Multipole Fields . . . 2.5.2 Multipole Strengths for a Non-Uniform Field Expansion . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Multipole Strengths for a Contoured Entrance or Exit Boundary of a Dipole . . . . . . . . . . . . 2.6 The Equations of Motion in Their Final Form to Second Order . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Description of the Trajectories and the Coefficients of the Taylor Expansion . . . . . . . . . . . . 10 11