 Photoelastic fringe order

The photoelastic fringe order $N$ expresses the magnitude of the full relative phase retardation between two principal optical components in cycles:For a uniform specimen obeying the linear stress-optic law, $N=t|{\sigma }_1-{\sigma }_2|/f_{\sigma }$, where $t$ is thickness and $f_{\sigma }$ is the photoelastic fringe value. The order is dimensionless and need not be an integer. One unit means one additional complete relative-phase cycle, not a separate stress component.

The connection to a visible fringe depends on the optical arrangement. For ideal crossed linear polarizers and a uniform retarder with optical axes at angle $\alpha$ to the incident polarization, the transmitted intensity satisfiesHere $I_0$ is the intensity entering the specimen after the first polarizer, and $\delta$ denotes the full relative phase. At $\alpha =45^{\circ }$, integer orders are dark and half-integer orders are bright. At axis alignment $\alpha =0$, the image is dark for every order. The crossed-polarizer derivation therefore separates retardation-related extinction from axis-alignment extinction.^[ref]

For an original ambiguity example at45°, orders0.25,0.75 and1.25 all give $I/I_0=1/2$. With $f_{\sigma }=12000N/m$ and $t=3\mathrm{mm}$, they correspond to stress-difference magnitudes1,3 and5MPa. Brightness alone does not choose among them. Referencing an unloaded state and tracking changes continuously during loading can supply information for photoelastic phase unwrapping.

The primary Illinois apparatus makes these controls concrete: a leadscrew loads the specimen, a load cell records force, monochromatic light and polarizing elements reveal the pattern, and a camera records it. Its magnitude-observation setting uses quarter-wave plates to remove the direction-related fringes.^[ref] That changes the observation geometry but does not remove the need to establish absolute order.

An order map under a known photoelastic fringe value gives principal-stress difference in the stated specimen model. It does not by itself give the two stresses individually or the sign of the stress-optic coefficient. Wavelength, initial retardation and the through-thickness stress model remain part of the interpretation.