Efficient algorithms for electromagnetic scattering in symmetric structures
Fahimeh Sepehripour defended her PhD thesis at the Department of Electrical Engineering on January 14th.
Numerical methods like the Method of Moments (MoM) are vital in electromagnetics for solving complex problems but face challenges with computational cost as problem sizes grow. In her PhD research Fahimeh Sepehripour addresses these challenges by exploiting the symmetry of rotationally symmetric objects, or bodies of revolution (BORs), reducing three-dimensional problems to two dimensions. To overcome difficulties posed by singular equations in MoM, novel algorithms using recurrence relations were developed for efficient and accurate computation. Validated against various BOR configurations, this approach advances applications in optical scatterometry, wafer metrology, and beyond, including the design of rotationally symmetric antennas.
Numerical methods are extensively used in electromagnetics, as they allow for the solution of complex problems that are difficult or even impossible to solve analytically. A prominent example is the method of moments (MOM), which provides an efficient way to solve complex electromagnetic problems by transforming the associated equations into a system of simplified linear equations. Despite these benefits, the numerical nature of such methods introduces certain challenges. In particular, as the size or complexity of the geometry under analysis grows, both memory requirements and solution times increase significantly. This can make solving large-scale or detailed problems computationally expensive and time-consuming.
Symmetry reduces computational burden
Fortunately, the presence of symmetry in many real problems can drastically reduce the mentioned computational burden. For instance, in electromagnetic problems involving rotationally symmetric objects (also known as bodies of revolutions), the computational domain can be reduced from three dimensions to two. Specifically, for such objects, the equations only need to be solved along the associated generating curve- a curve that when rotated around the axis of symmetry, defines the full three-dimensional shape. This feature reduces the computational load and makes the problem more manageable.
Even in cases where rotationally symmetric feature of the objects allow one to reduce the computational burden, the corresponding equations in the Method of Moments (MoM) can still be computationally demanding. This challenge arises because these equations can become ‘singular’, which can cause difficulties in finding a unique and stable solution for the problem. In simpler terms, the singular nature of such equations makes the resulting system difficult to solve directly, thereby increasing the computational load once again. Therefore, it is crucial to develop efficient techniques that specifically tackle these problematic singularities.
Introduction of appropriate recurrence relations
As part of this PhD research has proposed numerical algorithms which makes the computation of the aforementioned singular equations expedient and accurate. In particular, appropriate recurrence relations have been introduced to efficiently compute the kernel (core component) of the singular equations that arise when addressing the electromagnetic scattering by bodies of revolution. The effectiveness of the approach is verified by computing the scattering from BORs having various shapes and sizes and by comparing the results with the literature and other independent computational techniques.
Elimination of redundant computations
In optical scatterometry applications, bodies of revolution (BORs) are frequently utilized as isolation surfaces to divide the computational domain into subdomains. This approach helps to eliminate redundant computations when the features of a particular subdomain remain constant. The process of using BORs as separation surfaces in wafer metrology is conceptually illustrated in Figure 1 and consists of three main steps: (a) Analysis of scattering by BOR in a homogeneous medium, (b) coupling between BOR and grating, (c) coupling between BOR and product structure.
This PhD research demonstrates that the proposed numerical algorithm can effectively address all the aforementioned steps mentioned in the modeling process of the BOR in wafer metrology. This analysis is not only relevant for optical scatterometry applications, it also finds application in other areas of electromagnetics, for instance for the analysis of antennas with a rotationally symmetric shape.
Title of PhD thesis: . Promotor: Prof. Martijn van Beurden. Co-promotor: Dr. Bas de Hon.