典型分形结构光栅的Fraunhofer衍射数值模拟

NUMERICAL SIMULATION OF FRAUNHOFER DIFFRACTION OF TYPICAL FRACTAL STRUCTURE GRATINGS

本文旨在研究Cantor条幅、Cantor地毯、Sierpinski垫片三种自相似分形结构的Fraunhofer衍射场在屏上的光强分布以及自相似分形结构的几何规律.通过复数积分法和位移相移定理,得到了屏幕上不同位置相对光强的数学表达式。随后,使用MATLAB进行数值计算和后处理,以直观的方式呈现出衍射图案.结果显示:衍射光强分布主体对称性由母代衍射图样的对称性确定;随着分形子代的增多,衍射图样在母代对称结构图样上逐渐分支复杂,最终收敛于类母代衍射图样,并满足衍射反比律.内容为光栅衍射和位移相移定理的教学拓展提供参考案例。

This paper aims to investigate the Fraunhofer diffraction patterns on a screen produced by three self-similar fractal structures:Cantor stripes,Cantor carpet,and Sierpinski carpet.Additionally,it explores the geometric principles underlying these self-similar fractal structures.Mathematical expressions for the relative intensity of light at different positions on the screen are derived through complex integration and the displacement-phase theorem.Subsequently,numerical calculations and post-processing are performed using MATLAB to visually present the diffraction patterns.The results reveal that the overall symmetry of the diffraction intensity distribution is determined by the symmetry of the parent diffraction pattern.As the number of fractal offspring increases,the diffraction pattern gradually branches and becomes more complex on the symmetric structure pattern of the parent,eventually converging to the diffraction pattern similar to the parent,and satisfying the inverse diffraction law.The content provides a reference case for teaching extensions of the grating diffraction and displacement-phase shift theorems.