Abstract text (incl. figure legends and references)

Introduction

The theoretical formulation of charge particle optics typically concerns the systematic analysis of charge particle trajectories close to some design trajectory / optical axis in a device producing deliberately configured electrostatic and magnetostatic fields. Cornerstone results consider the expansion of path deviations into a hierarchical system of differential equations, which may be solved in an iterative manner to yield paraxial and subsequent aberration optical properties of the system under consideration [1]. Note, however, that considerable technical difficulties arise when analyzing high-order aberrations, e.g., those of curved system, relativistic systems of combined electrical and magnetic fields, which are related to complex expressions, exploitation of symmetries (e.g., symplectic), or numerical implementation.

Objectives

Here, we seek to develop a particularly compact solution to the charge particle optics program employing Hamiltonian formulation of mechanics. That includes phase space coordinates adapted to (curved) optical axis, systematic expansion into a hierarchy of first order differential equations, simplification of equation system by employing adapted physical gauges, and symplectic symmetries. Finally, we seek to provide compact analysis of archetypal elements such as multipoles or magnetic lenses to high aberration orders.

Methods

The charge particle optics problem is considered as the propagation of deviation vectors, between design particles moving along the optical axis of a given device and particles in close vicinity to that design particles, along the optical axis [2]. In the Hamiltonian formulation of classical mechanics, these vectors are defined in phase space, i.e., contain both spatial and momentum difference to the design particle. We employ a canonical transformation to transform this conventional phase space into an adapted extended phase space containing time and energy as additional dimensions, which also implies a reparameterization of the trajectory parameter from time to optical axis coordinate. The latter is particularly well suited for description of particle coordinates in planes perpendicular to the optical axis. We finally employ a modified physical gauge (i.e., related to hyperaxial gauge), which is particularly well adapted to the tube-like geometry of the particle pencils around (curved) optical axis. The gauge simplifies expressions considerably and leads to explicit expressions of the potentials in terms of integrals of the physical electrostatic and magnetostatic fields. Last but not least, we also discuss the numerical implementation of this method within the boundary element formalism.

Results

Employing the above formalism, we set up a compact system of first order differential equations in 6 dimensional extended phase space describing paraxial and aberrational properties of general (curved) optical systems in a hierarchical manner. The equations obey symplectic symmetries at each expansion level, leading, e.g., to interdependencies among error paths or aberration coefficients. We solve these equations for a number of typical optical devices, such as quadrupole and magnetic lens.

[1] P.W. Hawkes, E. Kasper, Principles of Electron Optics Vol. 1: Basic geometrical optics, Principles of Electron Optics.

[2] F. Kern, J. Krehl, A. Thampi, A. Lubk, Optik 242, 167242.

[3] We acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) (No. 461150024.)