基于辛几何理论的散射场焦散区计算方法的研究

In high-frequency conditions, high-frequency approximation methods are applied to solve the electromagnetic wave equation. These methods, which are often used, are geometrical optics (G.O.) approximation, geometrical theory of diffraction (G.T.D.), physical optics (P.O.) approximation and etc. The solutions of the electromagnetic wave equation in the non-caustic region obtained by the methods are satisfactory. But the ones in the caustic region are not ideal. Some methods are not valid in the caustic region. So, it is very necessary to develop a new method for finding the solutions in the caustic region. In mathematical point of view, there is a singularity in the caustics in physical space, but the singularity is not a real one. In fact, the solutions of the electromagnetic wave equation are not singular. It is because the primarily simple expression of G.O. approximation is not suitable for the caustic region that the solutions are singular. In this project, a new symplectic geometrical high-frequency approximation is used. We have made the following research work 1)Study on the geometrical high-frequency approximation theory. 2) Solution on electromagnetic wave propagation in a reflector by symplectic geometrical high-frequency approximation 3) Solving the propagation of electromagnetic wave in the inhomogeneous media Meanwhile, the method for extending the solutions obtained by to the shadow region is also presented. It proves to be satisfactory as well.The method deserves further research.

本课题主要采用辛几何理论的高频近似方法，求解凹面体散射中焦散区散射场的问题。在辛空间中建立几种典型具有焦散现象的模型，并在辛空间中计算出焦散区的场值。利用含参变量的付立叶逆变换把辛空间中焦散区的解变换到实际的物理空间中。本项目首次引入辛几何理论及辛算法来处理电磁场焦散区的问题，是一个值得探讨的新方法。