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An improved full-vector finite-difference (FD) complex mode solver for circularly symmetric optical waveguides is developed and validated. The formulations are derived from Taylor expansion of transverse electric and magnetic fields and match of boundary conditions at radial index discontinuities. Calculation of the guided and leaky modes of step index fibers, Bragg fibers, and surface plasmon polaritons structures shows significant improvement in terms of accuracy and rate of convergence without increase of computational efforts, in comparison with the conventional FD scheme based on the average-index approximation. It is demonstrated that the performance of the new scheme is robust for high-order modes and waveguides with high index contrast. Contrary to the conventional FD solvers that claim the magnetic field formulation is more rigorous than the electric field formulation, the improved solver yields practically identical results for both formulations.