# Team 1: Tensor tomography of stress-induced birefringence in commercial glasses

Wednesday, August 5, 2009 - 9:40am - 10:00am

EE/CS 3-180

Douglas Allan (Corning Incorporated)

href=/2008-2009/MM8.5-14.09/activities/Allan-Douglas/figure1.jpg>

href=/2008-2009/MM8.5-14.09/activities/Allan-Douglas/figure2.gif>

src=/2008-2009/MM8.5-14.09/activities/Allan-Douglas/figure2.gif>

*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

MSC Code:

74B05

Keywords: