Group-theoretic Approach To The Theory Of Crossed Gratings And Its Application To The Fourier Modal Method

Crossed gratings are gratings that are periodic in two noncollinear directions. With the increase of engineering demands and the development of rigorous vectorial diffraction theory, crossed gratings are playing more and more important roles in scientific research and applications. Although many rigorous numerical methods have been developed for modeling crossed gratings, the heavy computation burden of these methods has been the biggest obstacle that prevents their application. Therefore, it is one of the most imperative tasks for grating theoreticians to dramatically improve the computation efficiency of the existing methods.This thesis presents an approach to utilizing the structural symmetries of crossed gratings to simplify the solution of diffraction problems. A systematic investigation into this problem is carried out by use of group theory that is the most powerful mathematical tool for treating symmetry problems.The structural symmetries of crossed gratings are first cataloged by the aid of plane crystallography. Therefore, all crossed-grating patterns belong to seventeen plane groups that are the combinations of ten plane point groups and five plane lattices. Then the specific steps are deduced to determine the plane group of any given crossed-grating structure. Since the composite operations are not taken into account at present, only ten plane groups without composite operations are treated in the thesis.For the ten plane groups, a group-theoretic approach is established based on group representation theory. It decomposes any crossed-grating problem with"generalized symmetric incidence"into a superposition of several so-called symmetrical basis problems. Then with the symmetry relations of the fields (i.e., the symmetry modes of the grating), the symmetrical basis problems are solved with symmetry simplifications. After the superposition...