 Stress-optic law

The stress-optic law relates stress-induced birefringence to a mechanical stress difference. For an initially optically isotropic material in a suitable linear regime,where ${\sigma }_1,{\sigma }_2$ are the principal stresses normal to the light path and $C(\lambda )$ is the stress-optic coefficient. Its units are $\mathrm{P}{\mathrm{a}}^{-1}$, and its wavelength dependence matters when comparing optical measurements.^[ref]

For uniform thickness $t$ and uniform relevant stress directions along the path, the measured phase retardation isThe measurement determines a difference, not both stresses separately. States $(20,5)\mathrm{MPa}$ and $(40,25)\mathrm{MPa}$ have the same difference and therefore the same retardation in this model. Extra mechanical information is needed to separate their absolute stress components.

A uniaxial calibration specimen gives a practical coefficient measurement. With axial force $F$, known sectional area $A$ and negligible transverse stress, ${\sigma }_1-{\sigma }_2=F/A$. If retardation plotted against force has slope $s=d\delta /dF$, thenFor illustrative $s=0.003\mathrm{rad}{\mathrm{N}}^{-1}$, $\lambda =532\mathrm{nm}$, $A=20\mathrm{m}{\mathrm{m}}^2$ and $t=2\mathrm{mm}$, this gives $C\approx 2.54\times 10^{-12}\mathrm{P}{\mathrm{a}}^{-1}$. A $200\mathrm{N}$ load would give $10\mathrm{MPa}$ stress difference and $0.60$ radians retardation. These are chosen calibration numbers, not a universal coefficient for a plastic.

A polariscope with controlled loading, a force readout and polarization analysis supplies the relevant apparatus.^[ref] Background birefringence, thickness, temperature and wavelength should be accounted for. A measured intensity alone can have multiple compatible phase delays, and a changing optical-axis direction through a three-dimensional specimen can invalidate the simple scalar-path formula. The law is most useful when its optical and mechanical assumptions are checked together.