A short way of solving advanced problems in electromagnetic fields and other linear systems

A linear system is characterized in the abstract by a source vectora, a response vector a system matrixR, connected byα = Ra. Specific interpretations fora , given for electromagnetic fields, the scalar producta Rbbeing the reaction. The advantages of this method are illustrated by application to various problems. Mode expansions are defined by the eigenvector equationRm = Cₘm,Cₘbeing the eigenvalue andmthe mode source. This means the mode is generated by a combination of electric and magnetic sources which, apart from the constant multiplierCₘ, are everywhere equal to the electric and magnetic fields, respectively. Such mode expansions are applied to typical waveguide problems. Waveguide theory is set up in terms of unit voltage and unit current mode sources,vandi, wherev v = 1 = i i. Then the definitions of waveguide current and voltage coincide with those for circuit theory, e.g., the mode current at cross sectionPis the reaction on a unit voltage source placed atP. The method also greatly simplifies scattering and antenna problems, such as the pattern of a monopole antenna which is immersed in a layer of gyrotropic plasma.