Abstract
Accurate modeling of electrostatic deflector systems is necessary to improve the resolution of particle-optical devices such as electron beam lithography systems and ultrafast electron microscopes. Traditional models often neglect the fringing fields at the boundaries of the deflector plates, despite their critical influence on the transverse and longitudinal velocities of charged particles and, consequently, on their actual trajectories. This paper presents a comprehensive analytical study of the electrostatic field and capacitance of deflector plates equipped with parallel at the input and expanding grounded screens at the output. Such a configuration enables enhanced beam control while mitigating uncontrolled stray fields. Using methods from the theory of functions of complex variables, we derive exact expressions for the potential distribution in the system. These solutions allow one to calculate the spatially varying electrostatic field strength in the deflector and determine the capacitance of deflector plates of various configurations. The approach can be used even for very short deflector plates. The results confirm that expanding grounded screens significantly localize the fringing field as well as the effects of uncontrolled stray fields. This analytical model provides a robust basis for optimizing the control of charged particle beams and can be used in the design of modern experimental applications of electron and ion optics.