Stress-optic law
= Stress-optic law
{title2=$\Delta n=C(\lambda)(\sigma_1-\sigma_2)$}
The <stress-optic law> relates stress-induced <birefringence> to a mechanical stress difference. For an initially optically isotropic material in a suitable linear regime,
$$
\Delta n=C(\lambda)(\sigma_1-\sigma_2),
$$
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 $\mathrm{Pa^{-1}}$, and its wavelength dependence matters when comparing optical measurements.<Experimental Strain Analysis by David Roylance>[Printed p. 6, stress-optical law and coefficient]{ref}
For uniform thickness $t$ and uniform relevant stress directions along the path, the measured <phase retardation> is
$$
\delta=\frac{2\pi tC}{\lambda}(\sigma_1-\sigma_2).
$$
The 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 $s=d\delta/dF$, then
$$
C=\frac{s\lambda A}{2\pi t}.
$$
For illustrative $s=0.003\,\mathrm{rad\,N^{-1}}$, $\lambda=532\,\mathrm{nm}$, $A=20\,\mathrm{mm^2}$ and $t=2\,\mathrm{mm}$, this gives $C\approx2.54\times10^{-12}\,\mathrm{Pa^{-1}}$. 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.<Polariscope and Photoelasticity by the University of Illinois Mechanical Testing Instructional Laboratory>[Loading fixture, load cell and direction/magnitude settings]{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.