A modal impedance-angle formalism : rigorous proofs for optical fiber mode counting and bracketing

In a companion paper, a complex-power-flow variational scheme is applied to analyze mode propagation along open circularly cylindrical graded-index waveguides. It leads to a characteristic equation in terms of impedances rather than fields. The resulting impedance-angle formalism provides the basis for the full-wave generalization for optical fibers of the mode-counting scheme previously developed for a scalar wave propagation problem. The complex-power-flow variational scheme for bent waveguides is based on energy considerations. Hence, in its derivation, it is natural to consider a waveguide section (a volume) rather than a cross section (a surface). In the proof of the mode-counting and mode-bracketing theorems, the key issue is to show that the characteristic roots and the roots of the so-called separation function alternate. For general circularly cylindrical open waveguides, the required proofs are intricate. However, the special limiting cases in which the optical fiber is surrounded by electrically or magnetically perfectly conducting walls are tractable. To account for the general case, it appears to be necessary to regard a class of optical waveguide problems with a continuous transition from perfect electric conductor to perfect magnetic conductor boundary conditions via the situation pertaining to the actual exterior medium. Thus, a half-strip is constructed on which the so-called characteristic and separation graphs are seen to alternate. As spin-off, such a sweep might prove useful in the design of a fiber cladding