Development of an Undergraduate Diffraction Course

数学科学(21)这个项目开发适合数学、物理、化学、材料科学和电气工程的初高中学生的衍射课的文本材料和相关软件。这些材料在SIUC和其他一些学院和大学进行了课堂测试。在介绍了一些双变量傅里叶分析(包括菲涅尔核的傅里叶变换以及梳状函数、柯里函数和网格函数的傅里叶变换，它们推广了单变量采样函数)之后，本课程介绍了傅立叶光学的基本原理(菲涅耳衍射、成像、分辨率、光学傅里叶变换、分数傅立叶变换等)。并分析了与双变量“晶体”有关的远场衍射图，即RXR上的双周期函数。该软件允许学生为使用高分辨率胶片录像机打印在35 mm幻灯片上的灰度图像创建PostSCRIPT文件。当激光束穿过这些载玻片时，会产生相应的衍射图。来自数码相机、扫描仪和上述PostSCRIPT文件的图像在PC上进行处理，以说明课程中研究的各种现象。由此产生的数学(傅立叶分析、算符理论、墙纸群等)、实验(在昏暗的数学教室中使用廉价的激光进行)和模拟(使用快速PC和FFT)的综合提供了对现代应用数学的激动人心的介绍以及对光学、X射线结晶学和数字图像处理的深刻见解。

Mathematical Sciences (21)This project develops textual materials and related software for a diffraction course suitable for junior-senior level students from mathematics, physics, chemistry, materials science, and electrical engineering. The material was class tested at SIUC and a number of other colleges and universities. After introducing a bit of bivariate Fourier analysis (including Fourier transforms of the Fresnel kernel and of the comb, curry, and grid functions which generalize the univariate sampling function) the course presents the elements of Fourier optics (Fresnel diffraction, image formation, resolution, optical Fourier transforms, fractional Fourier transforms, etc.) and an analysis of the far field diffraction patterns associated with bivariate "crystals", i.e., doubly periodic functions on RXR . Software allows students to create PostScript files for greytone images that are printed on 35mm slides using a high-resolution film recorder. Passing a laser beam through these slides produces the corresponding diffraction patterns. Images from a digital camera, scanner, and the above mentioned PostScript files are processed on a PC to illustrate various phenomena studied during the course. The resulting synthesis of mathematics (Fourier analysis, operator theory, wallpaper groups, etc.), experimentation (done in a darkened mathematics classroom with an inexpensive laser), and simulation (using a fast PC and the FFT) provides a stimulating introduction to modern applied mathematics as well as deep insights into optics, X-ray crystallography, and digital image processing.