# 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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