Finding Taylor expansion of dispersion curve for arbitrarily indexed optical fibers by hyper-perturbation theory

After a conventional finite-element analysis for the propagation constants of guided modes in arbitrarily indexed optical fibers, a hyper-perturbation approach is proposed to determine the derivatives of propagation constants up to very high orders directly from the modal solutions. By considering the differentiation of the variational reaction formula, the approach develops a systematic algorithm to find higher order derivatives of the propagation constant and the modal solution from their lower order derivatives. The method involves numerical integration and requires only one eigenvalue evaluation, resulting in higher accuracy with less computational effort. Based on this approach, the explicit Taylor expansion formulas of the dispersion relation for step and parabolic-index optical fibers are presented.