Light propagation in weakly guiding optical fibres. A Laplacetransform approach

Abstract

In this paper we study the scalar Helmholtz equation, in cylindrical coordinates, for the propagation of light in a weakly guiding optical fibre with the refraction index sechR (R is the distance from the axis of the fibre). We accomplish this study with the help of the Laplace-transform method which proves to be particularly well suited to the purpose. The analysis around the singular points of the Laplace-transformed equation provides not only the asymptotic behaviour of the solution of the original equation (which is a general feature of the Laplace transformation) but can even be transferred to the region of regularity, more precisely around a point which is singular for the equation but regular for the solution. In principle, this makes the search of the eigenvalue in the Laplacetransformed space easier because the matching of the asymptotic solution with the solution in the left half-plane (in turn equivalent, in the original space, to the matching of the regular solution on the right of the origin with the asymptotic vanishing solution) is also «transferred» in the right half-plane, which is a regularity region in the above-mentioned sense. Cut-off frequencies (in particular for the second mode) arise at the same manner: a simple and general relation is established for them. We stop the analysis at the stage in which the asymptotic expansion of the Laplace transform around the point at infinity is still to be conveniently represented («summed»). However, the crude approximation of using only the first term of the expansion yields already encouraging results.