Rigorous asymptotic theory of evanescent waves for guided propagation

Choudhary and Felsen proposed a high-frequency asymptotic theory of ducted propagation based on the tracking of local evanescent plane waves. In their construction of the asymptotic expansions for the modal fields and eigenvalues they observed that the true modal fields must be analytic on the waveguide axis, and they imposed that condition on the modal amplitude coefficients, although the axis is not included in the domain of uniformity of the asymptotic expansion. Yet, the procedure yielded asymptotic solutions that agree completely with rigorously derived results for special refractive index profiles. The problem is examined here within the framework of the theory of the ordinary differential equation that describes the modal field. It is shown that the asymptotic modal expansion coefficients must be single valued within a strip of the complex coordinate plane, and that this condition yields the same results as the procedure of Choudhary and Felsen. These authors also sought to remedy the nonuniformity of the expansion near the axis by extracting the known exact modal solution for a parabolic refractive index. By constructing a uniform asymptotic representation, this local parabolic approximation is shown here to be valid near the axis, but its domain does not extend far enough to permit overlap with the above-noted asymptotic expansion away from the axis.