Uniaxial parallel-plate waveguide Green´s functions using a scalar potential formulation

Summary form only given. A scalar potential formulation is derived and subsequently used for the first time (to the author´s knowledge) to find the Green´s functions for a parallel-plate waveguide filled with an electrically and magnetically homogeneous uniaxial anisotropic medium. It is discussed that considerable simplification occurs in the potential based approach presented here and, in addition, significantly more hysical insight is gained due to the succinct and compact results. This simplicity and insight offered by a scalar potential formulation therefore constitutes a very important contribution and paves the way for future analysis of electromagnetic problems involving complex media. First, the scalar potential formulation for a magnetically and electrically uniaxial anisotropic medium is derived. It is assumed the medium is, in general, inhomogeneous along the longitudinal axis (i.e., the z-axis). This assumption is later relaxed to accommodate the homogeneous medium of interest here. The formulation is based upon the two-dimensional form of Helmholtz´s theorem and the identification of operator orthogonality. Next, expected and unexpected depolarizing dyads encountered in the derivation are identified and it is shown that the unexpected dyads are removable via complex plane analysis. This is vital since it provides the rigorous methodology for carefully handling the source point singularity and obtaining mathematically and physically consistent results. Next, scalar potential boundary conditions for perfect conducting parallel plates are derived. The potential-based Green´s functions are found using the superposition of the principal solution (i.e., solving the resulting forced scalar potential differential equations) and the scattered solution (i.e., solving the resulting unforced scalar potential wave equations) along with application of the derived parallel-plate scalar potential boundary conditions. Finally, the desired Green´s functions for th- electromagnetic fields are found via the field recovery process developed in the scalar potential formulation. Applications of the Green´s functions are discussed, including the nondestructive electromagnetic characterization of uniaxial media. A general comparison between potential and field-based approaches is also provided.