photoelasticimetry.md

@@ -21,11 +21,22 @@ this change is not completely blocked by the second circular polarizer and is pa
 through it, recorded by the camera.
 Photo-elasticity can be used to quantitatively measure local stress. Assuming
 that the relation between stress and refractive index is linear:
-$´n_1-n_2=C(\sigma_1-\sigma_2)´$
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+$`n_1-n_2=C(\sigma_1-\sigma_2)`$
+where $`\sigma_i`$ is an eigenvalue of the local stress tensor, n i is the refractive index for polar-
+ized light in the corresponding direction, and C is called the stress-optical coefficient.
+The relative phase shift of wave components in the eigen-directions of local stress
+tensor can be calculated using:
+$`\alpha=\fract{2\piCd}{\lambda}(\sigma_1-\sigma_2)`$
+where $`\lambda`$ is the wavelength of the light and d is the distance traveled inside the
+material. Given this phase shift, the intensity of the emergent wave is:
+$`I=I_0 sin^2(\frac{\alpha}{2})=sin^2(\fract{\piCd}{\lambda}(\sigma_1-\sigma_2)) `$
+Eq. 3.3 relates internal stress and intensity for a single wavelength. In the
+experiment, both the light and camera channels cover a wide range of wavelengths.
+Stress calculation is less noisy using narrow range of wavelength and single channel
+from the camera. However, it is still possible to retrieve stresses using white light.
+In such as case, stress-optical relation can be evaluated using empirical calibration.
+Here, we investigate how the photo-elastic response can be used to retrieve local
+stress in our experiment.