Computing Group Velocities and Group-Velocity Dispersions of Optical Fibers Through Automatic Differentiation of Explicit Forms of Propagation Constants

We present a method to compute differential coefficients of the first and second orders, for propagation constants of nondegenerate and degenerate modes, propagating in an optical fiber to calculate dispersion characteristics of group velocities (GV) and group-velocity dispersions (GVD). This method automatically differentiates explicit forms of eigenvalues extracted from determinants of a generalized eigenvalue problem. This problem is converted from a nonlinear eigenvalue problem derived from the Trefftz method (e.g., transfer matrix and multipole methods) by the Sakurai-Sugiura method. We compute the differential coefficients of the second order of degenerate modes without computing eigenvectors and their first order differential coefficients in the generalized eigenvalue problems. Therefore, in parametric optimization, we can compute the differential coefficients of all propagating modes without remodeling the waveguide cross-section, as analyzing the whole cross-section avoids the symmetric conditions derived from the numerical model. Computed results with the proposed method for step-index and holey fibers validate the method and its effectiveness. In particular, our results show that our method computes the GV and GVD more accurately than computation using numerical differentiation.