Optics Independent Project

A qualitative examination of birefringence and stress patterns in leucite

by Callan Ordoyne

Abstract

We used a birefringent material, leucite, to image the patterns of strain produced by stress at a point. We found the stress produced a distance-dependent shift in the index of refraction n of the stressed area, thus allowing us to image the stress over distance as a function of the phase shift of incident light.

Materials & Methods

We designed an experimental setup (see Figure 1) such that a Helium-Neon gas laser beam emitted through spatial filter (consisting of a short-focal-length lens that focused the beam through a pinhole) was linearly polarized by the first polarizer. We then placed the stressed object in the path of the beam, which then passed through a second linear polarizer placed at an orientation of 90° relative to the first polarizer. The beam then was projected onto an opaque ground glass plate, producing an image of the stressed object. This image on the plate thus served as the object for the lens which focused the image onto the roughly 4 square mm. capturing area of our CCD camera. Our stressed object was a small thin rectangular block of transparent, birefringent leucite. The properties of this piece of leucite are such that the axes of polarization are set in a fixed orientation, parallel to the lateral edges of the block. Thus, we placed the leucite block at a 45° angle to the crossed polarizers to ensure best possible transmittance of the laser beam. For the source of stress, the leucite was secured in a steel clamp, which effectively supplied stress at two points on opposite sides of the leucite block. Though stress was induced by both arms of the clamp, the image size of the camera only allowed one of the stressing points to be captured; all data was collected from the same single stress point on the leucite. For a control, the leucite was imaged without any induced stress. The clamp was then attached and stress applied by tightening the clamp. Images were captured by the camera at roughly equal intervals of increasing stress. However, the limits of our equipment did not allow us to quantitatively measure the amount of stress applied, and thus necessarily the judgment of "equal intervals of stress" was a qualitative one.

Results

The images of the stressed block of leucite captured at increasing levels of stress are shown in Figures 3-9, with the control in Figure 2. The camera used introduced a certain amount of noise into these images; the grainy appearance of the photos is due to the camera system. The horizontal striping seen in some of the pictures is also due to the camera's method of scanning in the data it receives. Figures 10 and 11 are the images I analyzed for a numeric analysis of how stress falls off with distance. I used Photoshop to sketch the central line of stress, and added points along the line at roughly the center of the rings. Both judgments were my own best estimates, and thus subject to a certain amount of human error. I then used the measurement tool in Photoshop to determine the distances from the point of stress to the center of each ring. Each transition from a light to a dark ring corresponds to a change in phase of ¹. This relationship is graphed in Figure 12 for both a medium level of stress and a higher level of stress.

Figure 2: No applied stress

Figures 3-9: Images of progressively increasing levels of stress.

Figures 10 & 11: The sketches used for determining distance between rings.

Discussion

We can interpret this set of images if we realize that brightness is correlated to change in phase, change in phase being the phase difference introduced by the differing indices of refraction of the material. Thus, a phase difference in even integer multiples of pi will result in the light being completely blocked by the second (crossed) polarizer. A phase difference of an odd integer multiple of pi will rotate the plane of polarization such that the light is completely transmitted through the second polarizer. Thus, in our images, each light and dark ring corresponds to an integer shift in pi. More precisely, intensity is equal to the square of initial energy of the incoming wave times the sine squared of half the change in phase. Thus, we can interpret the intensity rings in our photographs as changes in phase, with a transition from light to dark or vice versa corresponding to a phase shift of pi. Because the change in phase is a direct consequence of the change in indices of refraction, delta n, our measurement of the change in phase over distance is also a measurement of delta n over distance. In turn, delta n is a direct consequence of the applied stress (as the stress perturbs the material and thus shifts the indices of refraction). And so, by showing that the change in phase declines over distance from the point of stress, our graph also implies a sharp decrease in the level of stress over distance, which is consistent with the predictions of elasticity theory.

The next logical step in this project would be to obtain more quantitative data. In particular, it would be invaluable to have an accurate measurement of the strength of the force applied at any given time, as one could then gain a better understanding of exactly how the indices of refraction in the material change with the level of force applied. Further research could also include a similar analysis of different materials, and could apply a more rigorous understanding of elasticity theory than my current level of knowledge allows.

Acknowledgments

Mark Peterson, for cheerfully assisting at every level of this project and teaching me everything I know about the subject.

Ernie Provo, for finding all the equipment and miscellaneous stuff this project required.

MHC Physics department, for sponsoring this project and keeping the physics lounge supplied with cooca.