= Photoelastic fringe order
{title2=$N=|\delta|/(2\pi)$}
The <photoelastic fringe order> $N$ expresses the magnitude of the full relative <phase retardation> between two principal optical components in cycles:
$$
N=\frac{|\delta|}{2\pi}.
$$
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 satisfies
$$
\frac I{I_0}=\sin^2(2\alpha)\sin^2(\delta/2)=\sin^2(2\alpha)\sin^2(\pi N).
$$
Here $I_0$ 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.<Experimental Strain Analysis by David Roylance>[Printed pp.6–8, stress-optical law, analyzer transmission and extinction mechanisms]{ref}
For an original ambiguity example at45°, orders0.25,0.75 and1.25 all give $I/I_0=1/2$. With $f_\sigma=12000\,\mathrm{N/m}$ 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.<Polariscope and Photoelasticity by the University of Illinois Mechanical Testing Instructional Laboratory>[Loading and readout; direction/magnitude settings and quarter-wave plates]{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.
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