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.
