IMPROVEMENT OF FOURIER MODAL METHODS BY MEANS OF NON CONVENTIONAL COORDINATE SYSTEMS

摘要

Modal methods and mode matching techniques are well established methods to solve wave guide and scattering problems. One of their interesting features is that they easily allow to understand and give some physical insight in the physical phenomena. Such methods lead to an eigenvalue problem that is transformed into a numerical matrix eigenvalue problem by the method of moments. At this stage, the boundary conditions that the field of the physical problem have to satisfy are included in the chosen expansion basis. For periodic structures the Fourier basis is a natural one. From the numerical point of view one obtains very simple code. However the main drawback of most Fourier based methods is that they are not able to describe efficiently electromagnetic fields with sharp variations as is the case in nano structures that include metals. As a consequence, convergence is achieved with rather huge matrices. For 2D problems it does not matter, but what about 3D structures? In that case, the situation is not that simple and it is our opinion that there is a need for fast, reliable, and easy to implement algorithms. For structures with single or double negative materials, the Parametric Fourier Modal is one such method. The Fourier modal method has recently been improved with the idea of adaptive spatial resolution. By using a non-uniform sampling scheme that places more sampling points around the permittivity discontinuities, the total number of spatial harmonics required to achieve a given accuracy is significantly reduced. The idea of stretching thinner the space around the discontinuities is not entirely new in the larger context of rigorous numerical methods. What is new in our parametric approach as applied to gratings is that Maxwell's equations are written in the stretched coordinates under the covariant form. Thus the matrix operator takes into account the new information. The Aperiodic Fourier Modal is an extension of the classical Fourier Modal Method. In 1994 Berenger introduced the perfectly matched layers (PMLs) in finite-difference time-domain (FDTD). Since then, the PMLs have been successfully combined with others methods in particular in the frequency domain. In practice one main feature of the PMLs, which appears in many applications, lies in the fact that they allow to use modal expansion techniques. Hence, in the context of the Fourier Modal Method, PMLs introduce radiation condition while using periodic expansion functions. The paper is organized as follows: in the first two sections, we derive the Fourier Modal Method from Maxwell's equations written under the covariant form. In section 4, we give numerical examples that include adaptive spatial resolution and complex coordinates.