Hesam Mahmoudi Nezhad Neda, Ghaffarian Niasar Mohamad, Hagen Cornelis W, Kruit Pieter
Department of Imaging Physics, Delft University of Technology, Delft, The Netherlands.
Faculty of Electrical Engineering, DC Systems, Energy Conversion and Storage, Delft University of Technology, Delft, The Netherlands.
Sci Rep. 2024 Feb 28;14(1):4859. doi: 10.1038/s41598-024-55518-3.
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.
在电子光学中，电场计算在所有计算和模拟中都起着重要作用。诸如有限元法（FEM）、边界元法和有限差分法等精确的场计算方法已经使用多年。然而，这些方法的计算成本非常高，当尝试应用具有许多自由参数的静电透镜系统的自动化设计时，会使计算机模拟变得具有挑战性甚至不可行。因此，多年来，电子光学科学家一直在寻找一种快速且准确的场计算方法，以解决静电电子透镜系统设计和优化中的上述问题。本文提出了一种用于静电电子透镜系统中快速电场计算的新方法，该方法具有相当高的精度，使电子光学设计师能够在合理短的时间内设计和优化具有许多自由参数的静电透镜系统。该方法的本质是在轴对称坐标系中用轴向电势的四阶导数来表示离轴电势，并将其与该轴向位置处电极的电势相等。在有限数量的轴向位置上进行此操作，我们得到一组可以求解以获得轴向电势的方程，这对于计算透镜特性是必要的。我们将此方法称为四阶电极法，因为我们采用了高达四阶的轴向导数。为了求解这些方程，通过同时求解三组线性方程来计算轴向电势的五次样条近似。这些方程组是从拉普拉斯方程和描述五次样条的基本方程中提取的。将该方法的精度和速度与其他场计算方法，如有限元法和二阶电极法（SOEM）进行了比较。通过使用作者开发的基于遗传算法的静电透镜系统优化程序，在静电透镜系统的设计/优化中实现了新的场计算方法。将这种新的场计算方法在优化静电透镜系统光学参数方面的有效性与有限元法和二阶电极法进行了比较，并给出了结果。需要注意的是，该公式是针对一般轴对称静电电子透镜系统推导的，然而本文中所示的示例是关于圆柱形电极的，这是因为在软件中实现起来较为简单。
