Analysis of Some Mathematical Questions of Transformation Optics and Its Application to Stealth Carpet Design
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摘要: 变换光学(TO)是目前电磁领域的一个研究热点，为了对基于TO的隐身斗篷设计提供更深入的理论支持，该文对TO的3个基本数学问题进行了讨论。首先分析了基于Maxwell方程组3维坐标变换下变换形式的唯一性问题，提出了不同于参考文献的新变换方式，指出了可以灵活调整变换空间特性阻抗的变换方法。以此为基础，提出了以自由空间的场分布为原始空间，通过变换获取到可以隐藏在介质包围区域中并和介质阻抗匹配的隐身斗篷设计方法。然后以基于波动方程的2维TO为研究对象，分析了以电场分量波动方程为基础进行2维变换时，磁场所满足的变换关系和对应的边界条件特性，同样指出了可以隐藏在介质包围区域中并和介质阻抗匹配的隐身斗篷设计方法。最后从数学上证明了2维变换下共形变换对于设计出非各向异性介质隐身斗篷的充分必要性。以上述讨论为基础，该文给出了一个嵌入式斗篷的仿真验证示例。该文的研究内容和结论对基于TO的各种电磁应用提供了理论支持。Abstract: Transformation Optics (TO) is a hot topic in the research area of electrical-magnetic fields. For providing further theoretical support to the design of stealth carpet based on TO, three basic mathematic problems of TO are discussed in this paper. Firstly, the uniqueness of transformation form in three-dimensional transformation of Maxwell’s equations is analyzed. A new transformation model is proposed, which is different from the classical one shown in reference. The new model also leads to a new transformation method that can generate flexible characteristic impedance in transformation space. Based on this, a design method of stealth cloak or carpet that can be used to hide the target in an area surrounded by medium with given permittivity is discussed. During this process, only the field distribution in free space is required as the original field during mapping. Secondly, the two-dimensional transformation of the wave equation is studied. The transformation of the magnetic field component in the two-dimensional transformation based on the wave equation of the electric field component is analyzed. The boundary matching during transformation is also discussed. The two dimension design method of stealth cloak or carpet that can be used to hide a target in an area surrounded by medium with specified permittivity is also discussed. Finally, the sufficiency and necessity of conformal transformation for designing a two dimension stealth cloak with non-uniform and anisotropic medium are proved strictly. The simulation results of a stealth carpet embedded in material are given to verify the proposed method. The analysis and the related conclusion presented in the paper provide theoretical support to the related application based on TO.
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