In order to study propagation properties of hollow beams through a paraxial optical ABCD system with a hard-edged circle aperture and a lens,the expressions for the propagation of hollow Gaussian beams through such a system are derived based on the Collins diffraction integral.

Precise Cotes integration method was proposed by combining precise exponential matrix calculation and higher accurate Cotes integration.

And the new equation could be integrated step by step with precise exponential matrix calculation and Cotes integration.

Applying the Huygens-Fresnel diffractive integral,the on-axis intensity distribution and the phenomena of focal shift and focal switch are theoretical studied when a partially coherent annular light is focused by a lens.

Based on the generalized Huygens-Fresnel diffraction integral,the analytic propagation expressions for Lorentz beams passing through first-order axisymmetric optical system are derived.

Based on the misaligned diffraction integral formula for misaligned optical systems, the diffraction for misaligned Lohmann s fractional Fourier transformation system and quasi\|asymmetrically inhomogeneous medium are analysed, respectively.

The diffraction formula of the hollow Gaussian beam propagating through a circle and a misaligned 4f optical system is obtained by using the ABCD and Collins diffraction integral formula,and the properties of the diffraction field is studied.