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 electrostatic charged particle lenses involves changing many geometrical parameters of the lens electrodes as well as changing the voltages at each lens electrodes. The objective of the design is often to get the electrons passing through the lens system to be focused at a specific point and to minimize the aberrations of the lens. To make such a design is a laborious task for electron optical lens designers. A fully-automated optimization routine to relieve the laborious manual design of charged particle lens systems has been demanded for years, however, to achieve this outcome while many free optimization parameters are involved in the lens system design is quite a challenge. This is mainly due to calculations of the lens electric potential which are in general carried out with very time-consuming techniques that require meshing of the lens space. Currently it is not difficult to find the electron-optical software to conduct accurate field calculations such as EOD, GPT, CPO, Simion, etc., that can be used in an optimization loop. However, it can take months to get the results. For instance, the design of even a simple system using COMSOL takes such a long computational time that the designer might not have the patience to wait for the result (assuming the computational memory allows them to do so). Therefore, although some charged particle optics design programs exist which could change a few geometrical parameters of the lenses or the voltages (EOD, GPT, CPO, Simion, etc.), a fully-automated optimization routine which could make a design where all geometries and voltages of the lenses could be varied in a feasible time did not exist. A first attempt (SOEM) was made more than 30 years ago, but it still had too many limitations to be widely used. The main objective of this thesis therefore was to find a technique which would enable electron optical designers to tackle this problem...

The design of electrostatic electron lenses involves the choice of many geometrical parameters for the lens electrodes as well as the choice of voltages applied to the electrodes. The purpose of the design is to focus the electrons at a specific point and to minimize the aberrations of the lens. In a previous study, genetic algorithm optimization was introduced to aid the designer. For speeding up the electrostatic field calculations, new methods for analytical approximations of the field near the optical axis were introduced. In this paper, the influence of the main tuning parameters of the Genetic Algorithms is analyzed. The analysis is performed on a typical electrostatic lens systems including 6 electrodes. Different combinations of population sizes and number of generations are taken and the quality of the optimized lens system is compared. It is seen that within a constant computational effort (time or total number of system evaluations), a lower population size with a larger number of generations can achieve better results compared to having larger population size and fewer generations. The combination of Crossover Heuristic with Mutation Gaussian showed significantly better results compared to all other combinations of Mutations and Crossovers. Crossover Fraction is also evaluated to find the most suited values of this parameter.

To design electron lens systems, applying a fully automated optimization routine has not yet been feasible, especially for the case where the optimization has many free variables of the lens system, such as all parameters that define the geometry of the lens electrodes and the voltage of each electrode. Hence, the study of the implementation of different optimization procedures has not yet been possible either. In one of our previous studies, we have proposed to use the so-called Second Order Electrode Method (SOEM) which performs the electrostatic field calculations in a very short time by the approximations of the field near the optical axis. There, using SOEM in field calculation, a Genetic Algorithm (GA) was successfully implemented to optimize the electron lens systems. One of the questions that has not been studied and answered in the literature yet, is whether the GA is the most suitable option among different optimization techniques for the design/optimization of electron lens systems. In this paper, by implementing the SOEM technique as the field calculation method, different optimization procedures are implemented and their performances are compared. For this study, a typical six electrode lens system is employed. The implemented optimization techniques include calculus-based local optimization (‘Fmin’) and metaheuristic methods such as GA, Particle Swarm Optimization (PSO), and Simulated Annealing (SA). The results demonstrate that the population-based global optimization techniques like GA and PSO significantly outperform single-based local optimization methods such as ‘Fmin’ and SA. Additionally, PSO shows slightly better performance than GA, although it cannot be concluded that PSO will always outperform GA for every electron lens design problem. Furthermore, in the comparison between the two single-based optimization techniques, the metaheuristic approach (SA) outperforms the calculus-based one (‘Fmin’). Hence, we recommend implementing metaheuristic, global, population-based optimization techniques like GA and PSO for the optimization electron lens systems.

One of the major challenges in optical lens design is to ascertain the lens system with the highest image quality. The image quality of the lens system, which is a measure of the performance of the lens, is a function of aberrations. This function is highly nonlinear and leads to the presence of multiple local minima in the design (optimization) landscape. Evolutionary algorithms, specifically Genetic Algorithm, are receiving attention in this field as an efficient global optimization techniques for multi variables and nonlinear objective functions. However, to the best of our knowledge, studies are as yet unavailable to provide an analysis on the performance of GA and the influence of its tuning parameters on the optimization of these systems. Our research has been conducted to supply such information and to provide a guideline on using GA, in GA-aided optical lens designs. The performance of GA has been investigated in a general group of three-lens systems. It is shown that GA is an efficient optimization technique in this field, while applying the suitable tuning parameters of GA is crucial. It has been realized that Gaussian Mutation (Scale of 0.5), combined with Heuristic Crossover, with a Crossover Fraction of 0.6, was the option which yielded good (i.e. the challengeable practically expected) results. However, any variation of these parameters may prevent the system from ever reaching an optimal configuration.

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.

To optimize the design of a system of electrostatic lenses can be quite challenging. Especially when many lens electrodes are involved, the number of design parameters, such as electrode thickness, radius, gaps between electrodes and voltage, increases rapidly. Therefore, it would be really helpful when optimization routines can be used. There have been some attempts to develop optimization programs, such as Szilagy et al. [1] and Adriaanse et al. [2], but they used to be not very accurate. In the meantime, computers have become much more powerful, making it attractive to revisit the problem. In this work we apply a Genetic Algorithm [3] for the optimization, and MATLAB was chosen for coding.

In electron lens design, finding the optimum lens system for theapplication at hand, is still quite a challenge. The situation becomes especially more complicated when many lens electrodesare involved, because the number of free parameters of the optimization, such as electrode thickness, radii, gaps between electrodes and voltages, increases rapidly. Therefore, fast optimization routines are needed to tackle the problem. In the past, there have been some attempts to develop such optimization programs. Szilagy et al. [1] and Adriaanse et al. [2], have published someresults in 1989 on rough optimization of electrostatic lenses. However, using the above-mentioned methods, one could not get very accurate results. Now that we have more powerful computers and significantly better software, we revisit the problem. First we applied the so called “SOEM” (Second Order ElectrodeMethod) [2] for a fast (∼0.1sec) calculation of the axial potential. However, the results of the optimization were not accurate enough. To improve the accuracy of the SOEM-based optimization, we integrated a finite element based potential calculation method (using COMSOL). This way the potential calculation and the objective function calculation is more accurate, although the optimization becomes much slower. We propose a new approach that improves on the low speed of optimization while keeping the high accuracy results of the finite element method based potential calculation. This is done by first using a rough optimization based on the SOEM approach, resulting in a few approximately optimized systems. Then, using the parameters of the systems found, new sets of systems were defined using a small range of values around these parameters. Then the more accurate, COMSOL-based optimization was applied to this set of limited systems. We have tested our method on multi electrode systems up to 7 electrodes. We succeeded to very accurately optimize these systems within a few hours, with the electrode radii, gaps and voltages as free parameters, and the focus position as a constraint. [1] M.Szilagi. Yakowitz and M. Duff, Appl. Phys.Lett. 44, pp. 7-9, 1984. [2] J.P. Adriaanse, H.W.G Van der Steen and J.E. Barth, J.Vac. Sci. Technol. B7, pp. 651-666, 1989.