圆角矩形环的夫琅禾费衍射及其应用

针对圆角矩形环的夫琅禾费衍射精确解尚未获得的问题,建立圆角矩形环孔径的数学模型,基于基尔霍夫衍射积分公式,通过曲线坐标变换关系,利用傅里叶变换,得到圆角矩形环的夫琅禾费衍射精确解.作出该类环的衍射图样,并通过将圆角矩形环的夫琅禾费衍射推广到特例情况验证了研究结果的正确性.进一步分析表明,圆角矩形环衍射图在椭圆轮廓下出现了许多次级衍射峰,随着圆角矩形环的缺角因子减小或者环宽度增大,次级衍射峰特征逐步减弱,衍射图形趋于顺滑.最后,利用图形的可视化及特例推广说明了本研究在教学中的一些应用.

Fraunhofer Diffraction of Rounded Rectangle Ring and Its Application

To obtain the exact solution of fraunhofer diffraction of rounded rectangle ring, the mathematical model of rounded rectangle ring is established. Based on Kirchhoff''s diffraction integral formula, and through the curvilinear coordinate transformation and the Fourier transform, the fraunhofer diffraction exact solution of the rounded rectangle ring has been obtained. The diffraction pattern of this kind of ring is given and the validity of the results is illustrated by extending the fraunhofer diffraction of the rounded rectangle ring to the special case. Further analysis showed that the diffraction pattern of the rounded rectangular ring has many secondary diffraction peaks in the elliptical contour. With the decrease of angular factor or the increase of ring width, the characteristics of the secondary diffraction peak gradually weaken and the diffraction pattern tends to be smooth. Finally, its application in teaching is illustrated by the visualization of graphics and the extension of special cases.