Numerical field solution for an arbitrary asymmetrical graded-index planar waveguide

Abstract

We present a numerical solution for the field amplitudes in an asymmetrical graded-index planar waveguide with an arbitrary index profile. For small index differences between the surface and bulk indexes and for large differences between the cover and surface indexes, the modes of the diffused waveguides can be described entirely in terms of normalized mode index, diffusion depth, effective modal width, and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> number. The results show a high degree of accuracy when checked against exact published results for the parabolic and exponential index profiles. Universal charts for the modal fields in terms of the normalized quantities are presented for profiles of practical interest, namely, Gaussian and complimentary error function index profiles. We show that the modal width, while somewhat sensitive to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> number, is surprisingly insensitive to the index profile. Tailoring of the index profile, therefore, does not seem important for the efficient fiber-waveguide endfire coupling. Error involved in the estimation of the phase shift at the cover-surface boundary as function of the asymmetry parameter is shown to be rather small for practical cases. Also, the discrepancy in the field distribution between an almost symmetrical and a highly asymmetrical waveguide at the cover-surface boundary is illustrated with a practical example.