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In this work, we treat the rotation of the optical axis as a new degree offreedom and theoretically propose a tilted linear-crossing metamaterials(TLCMM). Specifically, the conical dispersions of the NLCMM and TLCMM have thesame shapes as type-I and type-II Dirac cones, respectively, in condensedmatter physics. Upon rotating the optical axis angle such that it is equal tothe cone angle, we find that this special TLCMM has the shape of a type-IIIDirac cone. This critical TLCMM can have many unique properties and is of afundamentally different nature than neighboring phases. When EM waves withlarge wave vectors are incident to a metamaterial with an open IFC to freespace, the incident EM wave is strongly reflected due to wave-vector mismatch.Here, we use boundary conditions and the causality law to reveal that TLCMMhigh-k modes can achieve abnormal refraction without reflection and filtering.Moreover, these phenomenon are observed experimentally in a planarcircuit-based system. The circuit-based TLCMM not only provides a versatileplatform for the study of robust negative refraction phenomena inmetamaterials, but also has a planar structure that is easier to integrate. Ourresults regarding the manipulation of EM waves may enable their use inplanar-integrated photonics including for directional propagation, cloaking,and switching.

Abnormal wave propagation of high-k modes in tilted linear-crossing metamaterials