Project Description: Birefringence refers to a different index of refraction for orthogonal light polarizations in a transparent material. In stress-free glasses (which are isotropic and can be made homogeneous) the birefringence is zero by symmetry. When such a glass is subjected to stress, even by squeezing with your fingers, stress-induced birefringence is readily observed. In real glasses a certain amount of stress is unavoidably frozen in during glass forming. It is of interest in a number of applications needing low or nearly zero birefringence to control and minimize the level of frozen-in stress birefringence. The goal of this project is to develop computational tools in Matlab to read limited sets of birefringence measurements and approximately reconstruct a stress distribution within the glass part that would be consistent with the measured birefringence scans. The general mathematical jargon for this procedure is "tensor tomography," but we are not trying to solve the problem at its most exact and sophisticated level. Instead we seek to make the absolutely simplest model for stresses within a sample that is approximately or adequately representative of the real stresses in the sample. Such an approximate reconstruction of stress would be useful to understand what stresses have developed in the sample and also how the birefringence would be altered if glass were removed, changing the stress boundary conditions. The model stress would have to obey the usual requirements of material continuity and force balance as well as the force-free boundary condition on the surface. Part of our goal is to achieve an adequate approximate description of stress using the fewest birefringence measurements possible. We have in mind a real-life application where reconstruction of the stress field from limited birefringence measurements would be useful. The application is in the manufacture of lens blanks, or blocks of extremely pure and highly homogeneous glass used to make the diffraction-limited optics for computer chip manufacture. Here the problem is fully three-dimensional, and at minimum several directions of birefringence measurement will be required. I am interested in possibly using Green function methods to solve for a stress distribution based on a set of initial strains. The strain field would constitute the unknown degrees of freedom for which we solve. This would automatically satisfy material continuity and force balance within the interior, and can be arranged also to satisfy the boundary conditions on faces. However, we may elect to pursue finite element methods or other choices depending on student interests and experience. References: Background on linear elastic theory and stress-induced birefringence can be found in many sources, including the web or textbooks in your university library. Note that we will work only in the linear regime and only with perfectly isotropic and homogeneous samples (when in their stress-free condition), so much of the mathematics is simplified. 1. One useful set of notes on linear elastic theory can be found at http://www.engin.brown.edu/courses/en224. See the Lecture Notes and especially the "Kelvin solution" of section 3.2 which is the basis of the Green function method. 2. Some basics of birefringence are included in the IMA Mathematical Modeling in Industry Workshop 2006 report found at http://www.ima.umn.edu/2005-2006/MM8.9-18.06/abstracts.html. See the link to the "Team 1 report" pdf . Prerequisites: Required: computing skills, numerical analysis skills, familiarity with Fourier analysis and convolution, ability to manipulate data arrays. Desired: some optics, some physics, familiarity with continuum elastic theory (stress and strain); the needed optical and glass-forming background will be supplied. Keywords:stress-induced birefringence, optical properties of glass, data analysis algorithms, tensor tomography, linear elastic theory