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Dielectric waveguides are important passive devices in optical communication systems. Circular-core fibers with slight ellipticity may lead to polarization-mode dispersion. A clear understanding of the propagation characteristics of the elliptical fibers thus becomes important for theoretical as well as practical purposes. Although mesh-dependent methods such as the finite-element method or finite-difference method, can be used to study such a complex structure, its computational task is very high. Strictly speaking, mesh-based solution does satisfy the Helmholtz equation and the solution only provided four to five significant digits. On the other hand, the highly accurate solution based on solving the Helmholtz equation of the elliptical coordinate system spend most its computational resources on computing the functional value and the zeros of the modified Mathieu functions of the first kind. Our method is based on linear combination of the exact mode-field solutions of the dielectric optical fiber. We apply the analytical continuity principle to obtain the simultaneous equation of the expansion coefficient vector. Since each basis solution satisfies the Helmholtz equation exactly, the overall solutions are very accurate and provide more than six significant digits for fibers with small elliptical eccentricity. In addition, only the Bessel functions are needed in our computation. Using cylindrical coordinate and symmetry, together with ACM principle, we simplify the problem of modal analysis of dielectric elliptical waveguides. This method also can be applied to some regular polygonal dielectric waveguides such as the large area VCESL.