 Stress-optic coefficient

The stress-optic coefficient $C$ measures the birefringence induced per unit principal-stress difference in the linear stress-optic law:Here $\Delta n$ is a difference of refractive indices, so $C$ has SI units ${\mathrm{Pa}}^{-1}$. The wavelength $\lambda$ matters: a coefficient determined with one illumination wavelength should not simply be reused with another. The signs of $C$ and $\Delta n$ also require a consistent ordering of the optical axes and stresses.^[ref]

For a uniform specimen of thickness $t$ viewed along a principal optical direction, the measured phase retardation isA known uniaxial load supplies $\Delta \sigma =F/A$ away from grips and other stress concentrations. Fitting the slope $s=d\delta /dF$ givesSubtracting the unloaded retardation removes a constant background, but a load-dependent background must be investigated rather than absorbed into the coefficient.

For an original illustrative measurement, let $\lambda =500\mathrm{nm}$, $t=2\mathrm{mm}$ and $\Delta \sigma =20\mathrm{MPa}$ produce $\delta =0.80\mathrm{rad}$. ThenDoubling the stress difference doubles the retardation only within the same linear regime. This numerical value is illustrative, not a material specification.

An equivalent magnitude convention uses the photoelastic fringe value $f_{\sigma }=\lambda /|C|$. One additional full retardation cycle corresponds to $|\Delta \sigma |=f_{\sigma }/t$ for the uniform model. A fringe magnitude alone does not determine the sign of $C$ or the separate principal stresses.

The Illinois primary apparatus description makes the measurement chain concrete: a leadscrew loads the specimen, a load cell measures force, monochromatic light passes through the specimen between polarizing elements, and a camera records fringes. Establishing $C$ connects those optical observations to a mechanical stress scale. Wavelength, thickness, unloaded birefringence and loading conditions belong with the reported coefficient.^[ref]