Lie Algebraic Methods in Charged Particle Optics

A Hamiltonian Mechanics Framework for Charge Particle Optics in Straight and Curved Systems

“…The above JVE (11) and the ensuing paraxial Newtonian equations of motion ( 13) treat both transversal and longitudinal deviations as well as straight and curvilinear systems on the same footing. Both describe paraxial optics completely and can be efficiently solved by numerical (step) solvers, which is the standard procedure, if the electric and magnetic fields vary along the optical axis.…”

“…The latter argument may be turned around, however, as gauge freedom can be sometimes exploited (A) to simplify the solution of the coupled systems of first-order differential equations ( 11) and (B) to simplify the relation between kinetic and canonical momentum in particular planes of interest (e.g., object and image plane). One particular useful trick is to fix the gauge such to render χ (t) time-independent (i.e., constant along the optical axis), which leads to a particularly simple integration of (11). Moreover, conservation of phase space is an additional structure available in the JVE, which can be incorporated into efficient approximation schemes (i.e., geometrical integration, see below).…”

“…Eq. ( 14) generally does not admit closed expressions for the transfer matrix and requires using numerical integrators (e.g., directly applied to (11)). Note, however, that closed solutions exist for a number of special cases; moreover, series expansions may be used to obtain analytical approximations of increasing accuracy.…”

“…of the Jacobian χ (t) matrix of the JVE (11) and reads by introducing a sharp jump in the transverse momentum at the entrance and exit plane. Indeed, the latter corresponds to the gauge transformation in the initial and final plane discussed further below.…”

“…The trajectory method iteratively solves Newton's equations of motion [1], the eikonal method [2][3][4][5] iterates Hamiltonian characteristic functions (here called the eikonal) with respect to trajectory deviations, and the Lie algebra method [6][7][8] is based on the canonical perturbation method in phase space [9,10]. A comprehensive comparison of these methods in electron optics has been carried out by T. Radlicka [11].…”