A polar coordinates N-axis finite-difference beam propagation method (FD-BPM) based on a novel step-length setting mode has been deducted,and an idea named variational polar coordinates FD-BPM is advanced in this paper,which will bring higher precision,more convenient computation and analysis. Comprehensive analysis to the rigorous two-dimensional (2D) circle waveguide is achieved. When the light wave is transmitted in the waveguide with different curvatures,a maximum ratio of conductible mode energy to initial energy existing is discovered,and short wave is easier to be formed conductible mode compared with in the same waveguide. Furthermore,investigation about S shape and C shape waveguide has been performed in depth. The propagation capability of C shape waveguide is more excellent than S shape waveguide,as the latter has a higher tangent loss,whereas this distinction reduces while the curvature increasing. Some available propagation rules of S and C shape waveguide with different curvatures are described. The transparence boundary condition (TBC) is thought as practicable while using the polar coordinates N-axis FD-BPM presented in this paper to research the rigorous 2D circle waveguide. So-called boundary echo disturbance caused by TBC,is proved inexistent.