TENSOR TOMOGRAPHY OF STRESS-INDUCED BIREFRINGENCE IN COMMERCIAL GLASSES

Abstract This paper describes the progress and results of a 10-day Work- shop on Mathematical Modeling in Industry XIII. Our team focused on the problem of understanding the stress state in a block of glass in- tended for use in making a photolithographic lens and relating those stresses to its birefringence. Our approach included studying both the “direct” problem, in which we derived a temperature distribution whose gradients approximates the stresses frozen into the glass during manufacturing, and the “inverse” problem, in which we seek to infer the internal stresses within the glass block based on measurements of birefringence along two axes. The latter is the “tensor tomogra- phy” problem mentioned in the title. Solution of the direct problem provided test data for studying the inverse problem. We chose to focus on an axisymmetric simplification since measured data show approximately axisymmetric optical properties. An important step toward explaining observed birefringence patterns was the solution of the steady-state temperature distribution in a cylindrical block of glass under slow cooling conditions. We exhibit that solution in this report and give details in an Appendix. 1 Contents