A modal impedance-angle formalism: Schemes for accurate graded-index bent-slab calculations and optical fiber mode counting

The propagation coefficients of guided modes along bent slabs and optical fibers may be calculated with the aid of a complex-power-flow, variational scheme. This scheme can be cast in the form of a Newton iterative scheme for a characteristic equation in terms of impedances rather than fields. Singularities of the associated Riccati equations for the impedances are circumvented via a generalized coordinate transformation, involving an angle for bent slabs, or a matrix of angles for optical fibers. Adopting the resulting impedance-angle formalism for bent slabs, numerical difficulties disappear, and accuracy ensues. The analysis of the vectorial optical-fiber problem benefits from its scalar bent-slab counterpart. In particular, the connection between the power flow and the impedance-angle formalism forms the physical underpinning for understanding that beyond a certain distance from the fiber core the derivative of the impedance matrix with respect to the propagation coefficient is positive definite. In turn, this provides the basis for the full-wave generalization for optical fibers of the mode-counting scheme developed in 1975 by Kuester and Chang for scalar wave propagation along a straight slab. With these key results, root-finding can be rendered more robust and efficient. A companion paper contains the necessary proofs for the complex-power-flow variational scheme and the mode-counting and mode-bracketing theorems.