In the last years, several numerical methods have been studied and applied to the analysis of high index contrast bent waveguides. Very often, the problem is treated using a conformal mapping, which translates the bending into an equivalent graded index profile and a straight waveguide. In this article, we discuss the implementation of a full vectorial 2D mode solver by means of the Aperiodic Fourier Modal Method, developed directly in cylindrical coordinates. This does not require the conformal mapping technique. In the first part of our work, we develop a shorthand notation and the mathematical rules useful to describe the problem in a matrix form. The calculation of propagation modes is then reconducted to the search of eigenvectors of a matrix. We will at first confront our formulation in 1D with results described in the literature. In a second time, we will use the complete 2D solver to determine the resonance frequencies and the quality factors of micro-ring resonators made on silicon surrounded by silica. These characteristics are indeed related to the real and imaginary parts of the propagation constants. By comparison with 3D-FDTD analysis, we will show that our implementation can be used to accurately describe the behavior of micro-rings having a bending radius as low as 1.1 µm in the near infrared region. This technique is general and can be applied to any micro-ring having an arbitrary cross-section and a quality factor which is less than 10000. Perspectives of this work include the study of the field propagation in a bent structure, as well as the coupling between micro-ring resonators and straight waveguides.