Wave and Optics Simulations

• G. Andrew Antonelli of Davidson College
• Wolfgang Christian of Davidson College
• Susan Fischer of Davidson College
• Robin Giles of Brandon University (Canada)
• Brian James of Salford University (United Kingdom)
• Ronald Stoner, Bowling Green University, (stoner@bgnet.bgsu.edu)

• DIFFRACT (Diffraction and Coherence), by Robin Giles, simulates some of the fundamental wave behaviors in Fresnel and Fraunhofer Diffraction, and other Interference and Coherence effects. In particular you will be able to study Fresnel diffraction phenomena associated with a point or set of points using the Huyghens construction and a slit or set of slits - where the transition from Fresnel to Fraunhofer will be established. You can also use a method developed by Cornu - the Cornu Spiral, and examine Fraunhofer diffraction with a single slit or set of slits, a rectangular aperture and a circular aperture. Finally, you can study Partial Coherence and fringe visibility in interference and diffraction observations. In the latter example you will be able to study the Michelson Stellar Interferometer and measure the separation distance in a double star and measure the diameter of single stars.

• SPECTRUM (Optical instruments), by Robin Giles, simulates the uses and modes of operation of four important optical instruments - the Diffraction Grating, the Prism Spectrometer, the Fabry-Perot Interferometer and the Michelson interferometer. You will look at the nature of the spectra, simulated interference patterns, and the question of resolving power.

• WAVE* (One-Dimensional Waves), by Wolfgang Christian, Andrew Antonelli and Susan Fischer, uses finite difference methods to study the time evolution of the following partial differential equations: classical wave, Schrödinger, diffusion, Klein-Gordon, sine-Gordon, phi four, and double sine-Gordon. The user may vary the initial function and boundary conditions. Unique features of the program include mouse driven drawing tools that enable the user to create sources, segments and detectors anywhere inside the medium. Double clicking on a segment allows the user to edit properties such as Ramsauer-Townsend effect in optics and quantum mechanics, respectively. Various types of analysis can be performed including detector value, space-time, Fourier analysis and energy density.

• CHAIN* (One-Dimensional Lattice of Coupled Oscillators), by Wolfgang Christian and Andrew Antonelli, allows the user to examine the time evolution of a one dimensional lattice of coupled oscillators. These oscillators are represented on screen as a chain of masses undergoing vertical displacement. The program allows the user to examine how the application of Newtonian mechanics to these masses leads to traveling and standing waves. The relationship between the lattice spacing and other properties such as dispersion, band gaps, and cut off frequency can be examined. Each mass can be assigned linear, quadratic and cubic nearest neighbor interactions as well as time dependent external force function. Global properties such as the total energy in the lattice or the Fourier transform of the lattice can be displayed as well as the time evolution of a single mass's dynamical variables.

• FOURIER (Fourier Analysis and Synthesis), written by Brian James, allows investigation of Fourier analysis and one and two dimensional Fourier transforms. In Fourier analysis users can choose from several predefined functions or enter their own functions either algebraically, numerically, or graphically. The build-up of a periodic function is illustrated as successive terms of the Fourier series are added in, and the effects of dispersion and attenuation on the evolution of the synthesized waveform can then be investigated. One and two dimensional discrete Fourier transforms can be produced for a range of standard and user-entered functions. The effects of filters on the inverse transforms are illustrated. The two dimensional transforms are shown as surface and contour plots. Image processing can be illustrated by filtering the transforms of gray level images so that when the inverse transforms are displayed it can be seen that the images have been modified.

• RAYTRACE (Ray Tracing and Lenses), by Brian James, lets the user explore the applications of ray tracing in geometrical optics. The fundamental principle of Fermat can be illustrated by plotting the path of a ray through two different materials between fixed points. The variation of the path of a ray through a region of changing refractive index can be used to investigate the formation of mirages. The variation of pulse delay in a fiber can be calculated as a function of its parameters and the characteristics of optical communication fibers are considered. The formation of primary and secondary rainbows due to dispersion of refractive index can be displayed. The matrix method of tracing rays through lenses can be used to investigate the images formed and show how aberrations in images arise and may be reduced.