De-Embedding Technique for Parameter Extraction of Biaxial Bianisotropic Metamaterial Slabs

Among material classes of isotropic, anisotropic, bi-isotropic, and bianisotropic, the latter class is important because, as different from other classes, the bianisotropic class can describe complex behavior of various materials through tensor expressions between electric and magnetic fields via electric and magnetic flux densities. All electromagnetic parameters (a total of seven) of a general bianisotropic material need to be accurately extracted for understanding its response and characteristic. Extraction methods can be utilized for such a goal. However, the extraction methods in the literature either require a formal calibration procedure or are limited to the extraction of three electromagnetic parameters only. In this study, an attractive self-calibrating waveguide method has been devised for determination of electromagnetic parameters of biaxial biansiotropic metamaterial (MM) slabs. Its first advantage is that it uses a thru connection in addition to the MM-loaded line connection (three different configurations - Case A, Case B, and Case C) and empty-line connection next to the MM-loaded line connection to extract all (seven) constitutive parameters together with the magnetoelectric coupling coefficient. Its second advantage is that it relies on explicit expressions and does not require any calibration process for such an extraction. Simulations and uncalibrated scattering parameter waveguide measurements at X-band ( $8.2 - 12.4$ GHz) were performed for a biaxial bianisotropic MM slab made by C-patterned resonators to validate the proposed method. It is observed that extracted electromagnetic parameters by the proposed method satisfy the passivity conditions over the entire frequency band except for a few frequency points or regions (e.g., $f = 8.2$ GHz). Its reason is expected to be associated with the difficulty in satisfying the continuous medium property at such frequencies. Besides, it is noted that two of the constitutive parameters ( $\varepsilon _{2}$ and $\mu _{3}$ ) illustrate a thickness-resonance behavior revealed by additional measurements of the Case B – MM structure with different lengths (7.675 mm, 9.21 mm, and 10.475 mm). Besides, it is noted that the resonance behavior of the Case C – MM structure changes with a change in periodicity. Specifically, when the periodicity in the x direction of this MM structure changes from approximately 3.26 mm to 2.29 mm, the resonance frequency shifts from nearly 10.43 GHz to 10.30 GHz (a 1.25% variation). Finally, sensitivity, uncertainty, and error analyses were also performed to evaluate and improve its accuracy. It is observed that $| \partial \varepsilon _{3} / \partial |\Lambda _{1A}| |$ is 14 times greater than $| \partial \xi _{0} / \partial |\Lambda _{1A}| |$ at 10.85 GHz where $\Lambda _{1A}$ is one of the terms related to measured S-parameters for the Case A – MM structure, and $|\star |$ denotes the magnitude of the complex number ‘ $\star $ ’.