{title2=$C(\lambda)$}

The <stress-optic coefficient> $C$ measures the <birefringence> induced per unit principal-stress difference in the linear <stress-optic law>:
$$
\Delta n=C(\lambda)(\sigma_1-\sigma_2).
$$
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.<Experimental Strain Analysis by David Roylance>[Printed p. 6, equation (5) and wavelength dependence]{ref}

For a uniform specimen of thickness $t$ viewed along a principal optical direction, the measured <phase retardation> is
$$
\delta=\frac{2\pi t}{\lambda}C(\lambda)\Delta\sigma.
$$
A known uniaxial load supplies $\Delta\sigma=F/A$ away from grips and other stress concentrations. Fitting the slope $s=d\delta/dF$ gives
$$
C=\frac{\lambda A}{2\pi t}s.
$$
Subtracting 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 $\delta=0.80\,\mathrm{rad}$. Then
$$
C=\frac{0.80(500\times10^{-9})}{2\pi(2\times10^{-3})(20\times10^6)}
 \approx1.59\times10^{-12}\,\mathrm{Pa}^{-1}.
$$
Doubling 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.<Polariscope and Photoelasticity by the University of Illinois Mechanical Testing Instructional Laboratory>[Specimen loading, monochromatic source, analyzer and camera]{ref}


 Stress-optic coefficient (source code)