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Photonic crystal fibers (PCFs) with many air holes and complicated geometries can be difficult to analyze using conventional waveguide mode solvers such as the finite element method. Boundary integral equation (BIE) methods are suitable for PCFs, since they formulate eigenvalue problems only on the interfaces and are capable of computing leaky modes accurately. To improve the efficiency, it is desirable to have high-order BIE methods that calculate the minimum number of functions on the interfaces. Existing BIE methods calculate two or four functions on the interfaces, but high-order implementations are only available for those with four functions. In this paper, a new high-order BIE method is developed and it calculates two functions on the interfaces. Numerical results indicate that the new BIE method achieves exponential convergence and extremely high accuracy.