We apply higher-order interface conditions to full-vectorial finite-difference mode analysis of circularly symmetric waveguides. It is very efficient in analyzing complex structures that analytic methods may face some difficulties in dealing with them, such as structures with absorption, leakage, multilayers, nonlinearity, and applications in beam propagation. A step-index optical fiber is modeled as an assessment, showing that three-, five-, and seven-point schemes yield second-, fourth-, and sixth-order convergence, respectively. In addition, the leakage of an antiresonant reflective terahertz pipe waveguide and an index anti-guided Bragg waveguide is calculated to demonstrate its performance. Since relatively coarse discretization can be adopted with higher-order formulation, it cannot only greatly save computation time and memory but also improve the ultimate accuracy.
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