Design Anisotropic Broadband epsilon-Near-Zero Metamaterials: Rigorous Use of Bergman and Milton Spectral Representations

A rigorous analytical approach is proposed based on the Bergman spectral representation and the Milton spectral representation of the effective permittivity for a composite system in order to design the anisotropic broadband epsilon-near-zero metamaterials. According to the Bergman and Milton spectral representations, the inverse problem for the anisotropic broadband epsilon-near-zero metamaterials is analytically established and solved by introducing the exactly proved identities. Meanwhile, the introduced identities theoretically reveal the identity of the Bergman and Milton spectral representations in mathematics and physics. With the precise solutions to the inverse problem, the conditions for designing the anisotropic broadband epsilon-near-zero metamaterials are clarified and demonstrated by the specific anisotropic broadband epsilon-near-zero meta-atoms of the lamination and the Hashin-Shtrikman coated-sphere microstructures through a theoretical analysis and a computational full-wave simulation.