The circular Airy beam is a type of autofocusing waves. The unique property of the circular Airy beam is that this beam can maintain quite low intensity profiles before the focal point and the maximum intensity of the beam can abruptly increase by orders of magnitude just at the focal point. The fine structures of the beam can be obtained by decreasing w. In this paper, firstly, the Fresnel, the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld diffraction theories were compared. It found that the Rayleigh-Sommerfeld diffraction theory was the most suitable method. Then, the propagation characteristics of the circular Airy beam with fine structures were studied in detail by using the Rayleigh-Sommerfeld diffraction theory. The results showed that the nonparaxial components of the field would increase as w decreases. At beginning, the increasing nonparaxial components enhanced the autofocus, resulting in an enhancement of the peak intensity and a decreasing of the full width at half-maximum at the focus. However, as w further decrease and beyond a certain value, more and more nonparaxial components became evanescent wave, and could not propagate to far field. The peak intensity at the focus decreased sharply.