This paper deals with theoretically the complicated index profile of an anisotropic fiber with an elliptical cladding. The mathematical steps taken in our analysis include the use of a series expansion of the index profile, the treatment of integral and derivative of a delta function and an iterative technique to obtain the Green functions of the optical field. Analytical expression of leaky losses of the fiber is given. Numerical examples show that the leaky losses of a fiber with an elliptical cladding are higher than that of an anisotropic step-index fiber, lower than that of a bow-tie fiber. With eccentricity increasing a single-mode single-polarization fiber is easier to make.