In electron optics, calculation of the electric field plays a major role in all computations and simulations. Accurate field calculation methods such as the finite element method (FEM), boundary element method and finite difference method, have been used for years. However, such methods are computationally very expensive and make the computer simulation challenging or even infeasible when trying to apply automated design of electrostatic lens systems with many free parameters. Hence, for years, electron optics scientists have been searching for a fast and accurate method of field calculation to tackle the aforementioned problem in the design and optimization of electrostatic electron lens systems. This paper presents a novel method for fast electric field calculation in electrostatic electron lens systems with reasonably high accuracy to enable the electron-optical designers to design and optimize an electrostatic lens system with many free parameters in a reasonably short time. The essence of the method is to express the off-axis potential in an axially symmetrical coordinate system in terms of derivatives of the axial potential up to the fourth order, and equate this to the potential of the electrode at that axial position. Doing this for a limited number of axial positions, we get a set of equations that can be solved to obtain the axial potential, necessary for calculating the lens properties. We name this method the fourth-order electrode method because we take the axial derivatives up to the fourth order. To solve the equations, a quintic spline approximation of the axial potential is calculated by solving three sets of linear equations simultaneously. The sets of equations are extracted from the Laplace equation and the fundamental equations that describe a quintic spline. The accuracy and speed of this method is compared with other field calculation methods, such as the FEM and second order electrode method (SOEM). The new field calculation method is implemented in design/optimization of electrostatic lens systems by using a genetic algorithm based optimization program for electrostatic lens systems developed by the authors. The effectiveness of this new field calculation method in optimizing optical parameters of electrostatic lens systems is compared with FEM and SOEM and the results are presented. It should be noted that the formulation is derived for general axis symmetrical electrostatic electron lens systems, however the examples shown in this paper are with cylindrical electrodes due to the simplicity of the implementation in the software.

The design of an electrostatic electron optical system with five electrodes and two objective functions is optimized using multiobjective genetic algorithms (MOGAs) optimization. The two objective functions considered are minimum probe size of the primary electron beam in a fixed image plane and maximum secondary electron detection efficiency at an in-lens detector plane. The time-consuming step is the calculation of the system potential. There are two methods to do this. The first is using COMSOL (finite element method) and the second is using the second-order electrode method (SOEM). The former makes the optimization process very slow but accurate, and the latter makes it fast but less accurate. A fully automated optimization strategy is presented, where a SOEM-based MOGA provides input systems for a COMSOL-based MOGA. This boosts the optimization process and reduces the optimization times by at least ∼10 times, from several days to a few hours. A typical optimized system has a probe size of 11.9 nm and a secondary electron detection efficiency of 80%. This new method can be implemented in electrostatic lens design with one or more objective functions and multiple free variables as a very efficient, fully automated optimization technique.

In electron optics, the design of electron lens systems is still a challenge. To optimize such systems, the objective function which should be calculated, depends on the electric potential distribution in the space created by the lenses. To obtain the electric potential, the existing methods are generally based on some mathematical techniques which need to mesh the space of the lens system and derive the electric potential at all mesh points. Hence, calculation of the objective function for such systems are computationally expensive. Therefore, applying a fully automatic optimization routine has not yet been feasible, especially for lens systems with many free variables. Hence, the study of objective-function landscape of such problems has not yet been performed. One of the questions of interest for optical designers, that has not been studied in the literature, is whether this problem can be solved by a local optimizer or is it necessary to apply a global optimizer. Recently we succeeded in implementing a method (based on a so-called SOEM (Second Order Electrode Method) technique) which calculates the electric potential in a fast and reasonably accurate way. In this paper, that method, is implemented to perform the study of local versus global optimization for electron lens design. The global optimization method here is performed by GA (Genetic Algorithm). The objective function is taken to be the probe size of the electron beams at the image plane. The results of our study show that the objective function of this problem has many local minima and the optimization of such problems cannot be handled by a local optimizer. GA is shown to perform well by overcoming these multiple-local minima to arrive at a global minima.

In electrostatic charged particle lens design, optimization of a multi-electrode lens with many free optimization parameters is still quite a challenge. A fully automated optimization routine is not yet available, mainly because the lens potential calculations are often done with very time-consuming methods that require meshing of the lens space. A new method is proposed that improves on the low speed of the potential calculation while keeping the high accuracy of the mesh-based calculation methods. This is done by first using a fast potential calculation based on the so-called Second-Order Electrode Method (SOEM), at the cost of losing some accuracy, and then using a Genetic Algorithm (GA) for the optimization. Then, by using the parameters of the approximate systems found from this optimization based on SOEM, an accurate GA optimization routine is performed based on potential calculation with the commercial finite element package COMSOL. A six-electrode electrostatic lens was optimized accurately within a few hours, using all lens dimensions and electrode voltages as free parameters and the focus position and maximum allowable electric fields between electrodes as constraints.