Comparison of six, four and two-vector formalisms for complex media

Advances in rapid prototyping using conductive-inkjet and 3D printing, for example, have made the fabrication of antennas and artificial complex media possible. These advances have prompted the need for alternative mathematical methods of analysis that are capable of handling anisotropic and bianisotropic media and also aid in better understanding of the physical nature of electromagnetic fields in complex media environments. As an example, a compact six-vector formalism for Maxwell´s equations was developed (I.V. Lindell, A.H. Sihvola, and K. Suchy, J. of Electrom. Waves and Appl., 9, 7/8, 887–903, 1995) that aids in the mathematical manipulation and subsequent solution of problems involving complex media. This formulation can accommodate fully populated material tensors (i.e., all material tensor elements are non-zero), however, the analysis leads to a block 3×3 matrix that requires inversion. If the material property tensors of permittivity and permeability have no transverse-longitudinal or longitudinal-transverse components (i.e., the xz, yz, zx, zy elements are zero), then a four-vector approach can be advantageously employed leading to a block 2×2 matrix which is more easily inverted due to the reduced dimensionality compared to the six-vector approach. In addition, a more compact and physically-insightful form is often achieved for this specialized case of material tensors. If, furthermore, the material property tensors are restricted to gyrotropic media, then a two-vector technique can be readily sought that results in a block 1×1 matrix formulation, which substantially simplifies the analysis and leads to enhanced clarity of the physical nature of the electromagnetic fields.