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Derivation, Cross-Validation, and Comparison of Analytic Formulas for Electrostatic Deflector Aberrations

Abstract

We derived first and second order analytic aberration formulas in the horizontal transverse plane for an electrostatic deflector specified by the reference orbit radius, the central angle spanning the deflector, and its inhomogeneity coefficients. The derivation was performed using an iterative order-by-order perturbation method in a Frenet-Serret beamline coordinate system. We produced a C program edabrt for calculation of the first and second order aberrations using the formulas that we derived.

The electrostatic deflector aberration formulas from Wollnik (1965)* disagree with COSY INFINITY and the aberration formulas that we derived here, and they also deviate from the first and second order symplecticity conditions. The reason for this discrepancy is that the aberration formulas from Wollnik (1965) consider only motion in the main field of the deflector and do not properly account for fringe field effects, in particular the necessarily occurring change of kinetic energy due to the change in potential. The code GIOS uses electrostatic aberration formulas from Wollnik (1965) and exhibits the same discrepancy.

We performed a comparison of first and second order electrostatic deflector aberrations for (1) the analytic formulas derived in this work, (2) differential-algebraic (DA) numerical integration of the equations of motion using the code COSY INFINITY, (3) the aberration formulas from Wollnik (1965), (4) aberrations computed using the code GIOS, and (5) the aberration formulas from Wollnik (1965) adjusted to account for the occurring change in potential. An electrostatic spherical deflector and an electrostatic cylindrical deflector were used as test cases for this comparison. There is excellent agreement between methods (1), (2), and (5), and these three methods satisfy the first and second order symplecticity conditions.

* H. Wollnik, Second Order Approximation of the Three-Dimensional Trajectories of Charged Particles in Deflecting Electrostatic and Magnetic Fields, Nucl. Instrum. Methods 34, 213 (1965).

E. Valetov, M. Berz, Advances in Imaging and Electron Physics, 213 (2020) 145-203. DOI: 10.1016/bs.aiep.2019.11.007

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