将夫琅和费衍射积分中的贝塞尔函数用高斯函数近似,得到经圆孔径准直光学系统衍射后高斯光束远场发散角的近似解析式.在不同衍射情况下,将其与严格夫琅禾费衍射积分进行比较,发现二者求解出的远场发散度接近.衍射孔径大小相同情况下,近似解析式与真实值的误差随准直前光束初始束腰的增大而减小;初始束腰不变的情况下,随着衍射孔径的增大,误差值略有上升,但最终趋于平稳.在初始束腰半径不小于2 μm的前提下,误差值最大不超过3.4%.该近似解析式在各种衍射情况下都能较为准确地表征准直圆孔衍射高斯光束远场发散度,且形式简单.对比不同衍射孔径和光束初始束腰条件下的光束发散度仿真结果可知,光束发散度随衍射孔径的增大而减小,随初始束腰的增大而增加.

By the aid of using Gaussian function instead of Bessel function in the Fraunhofer diffraction formula,the simple approximated analytic expression for the far-field divergence angle of Gaussian beam diffracted by a circular-aperture collimation optical system was obtained. For different diffraction conditions,the expression was compared by strict Fraunhofer diffraction expression,and the far-field divergences solved by these two expressions were nearly identical. With the same diffraction aperture,the error between approximated analytic solution and true value is reduced with the increase of the initial beam wrist;with the same initial beam wrist,the error increases slightly with the rise of the diffraction aperture semi-diameter,and finally flatten out. On condition that the intial waist is not less than 2 μm,the maximum of the error is no more than 3.4%. Therefore the approximated analytic expression correctly describes far-field diffraction divergence in every diffraction case,and has handy form. At the same time,through the comparision for the beam divergence in different diffraction aperture diameters and initial beam wrists,it can be seen that the beam divergence decreases with the increasing of the circular aperture semi-diameter and increases with the increasing of initial wrist.