A Vector Transform Solution Procedure for Solving Electromagnetic Problems in Cartesian Coordinates

Some vector transforms of importance to solving electromagnetic problems were introduced by Chew and Habashy in 1986. The primary advantage gained in the use of these transforms is that the analytical development of field solutions becomes streamlined by what amounts to a very powerful bookkeeping method. The aforementioned paper presents transforms for Cartesian, cylindrical, and elliptical geometries. A later paper by Weiss and Kahn demonstrated that these transforms emerge from careful consideration of the longitudinal components of the electric and magnetic fields. Applying the techniques to the Cartesian geometry, one finds a variation to the vector transform postulated by Chew and Habashy. This letter will discuss the analytical formulation of the variation and demonstrate its usefulness in the development of Green´s functions for a planar stratified medium. The foundation of this variation to the well-established development in Chew and Habashy´s work lies in obtaining it solely from consideration of the longitudinal electric and magnetic field components. This method then simplifies the derivations and necessary bookkeeping. Although demonstrated here for a simple geometry, the approach applies to multiple layers as effectively.