Exact eigenvalue equations for weakly guiding optical fibers with arbitrary graded-index profile

A modified analytical transfer matrix method (ATMM) is extended to investigate the LP_lm modes of weakly guiding optical fibers with arbitrary graded-index profile. Two classes of eigenvalue equation with respect to azimuthal number l = 0 and l ≠ 0 are developed, respectively. It is confirmed that the phase shift at a turning point in optical fibers is equal to π as well as in planar optical waveguides. The existence of the phase shift at the center of the core in optical fibers, which is equal to 2arctan2 for the case of l = 0, is also demonstrated. Moreover, a numerical comparison with the available reference solution shows that the proposed method is exact.