Intensity Method for measuring of stress in glass

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Stress in a bar as seen with polarisers In the picture above we see an example of the stress visualised with two crossed polarisers. We can estimate the stress by counting the "fringes": when the retardation is equal to 0 +/- 2p, the intensity is zero. So when the intensity goes 8 times to zero, the phase-shift = 8 * 2p .
In the picture above the material has a high Strain Optical Coefficient (SOC), so many fringes can be seen. Glass has a low SOC, so we have to interpolate between fringes. By eye this is not possible.

Set-up for measuring stress in glass with polarisers
After polariser at 45 degree:
Vector Ep is equal to Ex + Ey
Ex = Ey = sin( w t )
Vector Ep is equal to vector Ex + vector Ey
Afer the sample, difference is
Ey = sin( w t )
Ex = sin( w t + f )
Phase difference between Ex and Ey
Only the vectors normal to Ep pass the crossed polariser
Ec = ( sin(wt+f ) - sin(wt) ) /sqrt2  Decomposition of Ex and Ey in direction of Ec and Ep, vectors in direction Ep are blocked by crossed polariser
After the Analyser the intensity Ic for crossed polarisers is proportional to the square of Ec,
Ic = 1/2 Io ( sin(wt+f) - sin(wt) ) 2 = 1/2 Io ( 1 - cos( f ) ) (3)

Measuring the intensity is with the basic set-up as given above. This intensity is measured with a camera and a computer. To calculate the phase-shift we use formula (3). In this formula we have a constant Io which we have to eliminate. This is done with a second measurement, for instance done with the Analyser rotated over 90 degrees, so the two polarisers are parrallel. From the two measurements we can compute the phase-shift f :

Crossed polarisers Ic = 1/2 Io ( 1 - cos( f ) ) (3)
Parrallel polarisers Ip = 1/2 Io ( 1 + cos( f ) ) (4)
Elimination of Io Icn = ( Ip - Ic ) / ( Ip + Ic ) = cos( f ) (5)
Calculation of phase-shift f = arccos( Icn ) (6)

With SAMS we use this principle to measure stress in glass.