Complete Planar Impedance Conservation and PML Boundary Value Solutions

This paper shows that a straightforward way to evaluate planar impedance conservation boundary values of arbitrary boundary operators and orientations is to directly copy the construction from the well-understood 1-D Robin impedance-matching solution and then let the boundary rotate freely. Copying from this classical solution, hence, requires no use of split field and any of its stretched coordinate reformulations. In fact, it actually gives the complete general superposition solution, which also explains that the existing PML considers only a rational boundary operator and solves only a coordinately aligned example. The construction thus embraces both general order polynomial and nonpolynomial boundary operators and noncoordinately aligned boundary solutions to be put on top of the PML orthogonal terminations. A constructive source-field energy coupling relation is also introduced to simplify the impedance conservation analysis to read as the triple product identity. Evident by the various forms of Maxwell equations, we elaborate in detail that boundary value equations can also be written in various equivalent and endless redundant forms.