Appropriate formulation of the Characteristic equation for open nonreciprocal Layered waveguides with different upper and lower half-spaces

This paper will study the most convenient way of formulating the characteristic equation for shielded, parallel-plate, grounded, and open reciprocal/nonreciprocal planar layered waveguides, including the possibility of different upper and lower half-spaces. A detailed study of the suitable mapping for each kind of waveguide will lead to the formulation of analytic characteristic equations for shielded/parallel-plate/grounded reciprocal/nonreciprocal waveguides and also for open reciprocal ones. Although no mapping has been found to remove all the branch points of the characteristic equation for open nonreciprocal waveguides with different half-spaces, a robust approach will be proposed to overcome the main drawbacks caused by the multivalued nature of this problem. The combination of the suitable formulation of the characteristic equation with a systematic integral-nature root-searching strategy bears a reliable and efficient method. Some novel numerical results will be presented for open anisotropic reciprocal waveguides and for open magnetized ferrite slabs to illustrate the performance of the present proposal.