Uniqueness theorem and uniqueness of inverse problems for lossy anisotropic inhomogeneous structures with diagonal material tensors

Abstract

The uniqueness theorem for lossy anisotropic inhomogeneous structures with diagonal material tensors is proven. For these materials, we prove that all the elements of the constitutive tensors must be lossy. Materials like cloaks and lenses designed based on transformation-optics (TO) could be examples of such materials. The uniqueness theorem is about the uniqueness of Maxwell's equations solutions for particular sets of boundary conditions. We prove the uniqueness theorem for three cases: Single medium, media composed of two lossy anisotropic inhomogeneous materials with diagonal constitutive parameters, and media composed of two materials, where a material with diagonal material tensors is surrounded by an isotropic material. The latter case can be considered for the TO-based materials like invisibility cloaks or hyper-lenses that usually have diagonal anisotropic inhomogeneous constitutive parameters and also because cloaks or hyper-lenses are usually surrounded by free space and the sources are usually outside. For the sake of our argument in the uniqueness theorem that loss is the main condition for the validity of this theory, for cloaks as an example case of our analysis, it is analytically and numerically proven that the ideal invisibility phenomenon is possible for a simple lossy structure. We also examine the uniqueness of the inverse problem for such structures. We prove that all these materials have the same surface field distribution on a surface enclosing the area of interest, while solutions to Maxwell's equations inside them are different. The uniqueness of the inverse problem suggests that within the surface, the same medium should exactly be present. However, for lossy anisotropic inhomogeneous structures with diagonal constitutive parameters, this paper illustrates that this might not be true, despite the result of a previous study that shows that uniqueness could be true for some anisotropic inhomogeneous structures. For the analysis, the transverse electric Z-polarization is used. The simulation results are obtained by using a commercial Finite-Element based simulator.

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