Use of the orthogonal system of Laguerre-Gaussian functions in the theory of circularly symmetric optical waveguides

Using the Ritz method and the reference solution of the scalar wave equation for the infinitely extended parabolic-index profile (Laguerre-Gaussian functions), we calculate the field distributions and propagation constants for weakly guiding fibers having arbitrary circularly symmetric index profiles. We show that the solution reduces to a homogeneous real-symmetric matrix-eigenvalue problem, whose eigenvalues give the propagation constant of the desired mode. We present results for infinitely extended alpha profiles, for a parabolic profile with uniform cladding, and for a parabolic profile with an axial symmetric dip or hump.