Principle of Measuring Stress in Glass

Birefringence

Polarisers Set-up

In the glass arises a phase shift between light polarised in X or in Y direction
Ex = Ey A phase difference is built up Difference in phase
Ey = sin( w t )
Ex = sin( w t + f )

The stress in glass can be measured due to the birefringence of glass under stress. Birefringence means that a material has different indexes of refraction in different directions. Examples of minerals that have different indexes of refraction for the different crystal axes are calcite, mica and quartz.
In glass normally there is only one index of refraction. But when a force is acting on the glass, the glass is deformed; the elongation is proportional with the stress. The index of refraction n is changed and the change Dn is proportional with the applied stress s.
What is measured is the difference in optical path L caused by the difference Dn due to a difference in stress in (for instance) X and Y direction. This difference is built up during the (optical) path in the glass d in the Z direction:

L = D n . d [nm]       or       L = (sx sy) * SOC . d [ nm ]
with
L = (over the thickness of the sample) integrated difference in optical path [nm]
D n = difference in index of refraction in X and Y [-]
d = thickness of the sample [cm]
sx sy = difference in main stresses in X and Y directions [N/mm2]
SOC = Strain Optical Coefficient [(nm/cm)/(N/mm2)]

What is measured is the difference in phase between outgoing light vibrating parallel to the X or to the Y direction. So with the given relation between this phase shift f and the difference in optical path in (1), the stress in the sample can be computed with (2):

f = 2 p (L / l ) [rad] (1)
 
sx sy = ( f / ( 2 p ) ) . ( l / d ) / SOC (2)
 
with
f = phase difference between light polarized in X and in Y directions [rad]
l = wavelength of used light [nm]

Birefringence

Polarisers

Set-up
To see the difference in phase after the sample, the initial phase in X and Y direction must be the same. To get this, the simplest way is to use the same wave and to decompose it in the two directions X and Y. So a wave vibrating under 45 degrees with the X and Y direction is decomposed in 2 equal waves, one vibrating in the X direction and one vibrating in the Y direction.
A polariser placed before the sample is used to get only waves vibrating under 45 degrees with the X and Y directions.
A second polariser placed after the sample (often called Analyser) is used to compose the two directions again so we can see the difference. Due to the Analyser some light is blocked, this is depending on the phase shift in the sample.
Set-up for measuring stress in glass with polarisers
After the first polariser the
vector Ep is at 45 degree with the X-axis; decomposition:
Vector Ep is equal to Ex + Ey
Ex = Ey = sin( w t )
Vector Ep is equal to vector Ex + vector Ey
A phase difference is built up
in the sample, difference is
Ey = sin( w t )
Ex = sin( w t + f )
 
Phase difference between Ex and Ey
Ex and Ey can be decomposed in vectors parrallel to Ep and normal to Ep. 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)


Birefringence Polarisers

Set-up for measuring stress

All set-up's for measuring stress in glass consist of a light source and two polarisers, one before and one after the sample (called analyser). By this the stress can be seen by eye as a change in light intensity, as given in formula (3). To measure the stress, some measuring equipement can be added.

Set-up for measuring stress in glass
Light
Source
First polariser to get polarised light at 45 degrees with the main stress axis Sample which gives difference in phase between light polarised in X or in Y direction (Optional)
Measuring
Device
Second polariser (called Analyser) to get a light intensity dependent of the phase shift at the sample

Primary goal of this measuring equipment is to get a light intensity equal to zero, because this is easy to see by eye. A gray level equal to formula (3) is not so easy to measure by eye.


Various methods to measure stress and different measuring devices can be used.

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