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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.

The photoelastic fringe value $f_{\sigma }$ is the material-and-wavelength scale connecting photoelastic fringe order to principal-stress difference times specimen thickness in the linear stress-optic law. With vacuum wavelength $\lambda$ and stress-optic coefficient $C$, use the magnitude conventionHere $N=|\delta |/(2\pi )$ counts full relative-phase cycles and $t$ is specimen thickness. Since $C$ has units $\mathrm{P}{\mathrm{a}}^{-1}$, $f_{\sigma }$ has units $\mathrm{Pa}\mathrm{m}=N/m$. A value expressed in $\mathrm{MPa}\mathrm{mm}$ equals the same numerical value in $kN/m$. The wavelength dependence of $C$ must be retained.^[ref]

For an original example, take $\lambda =600\mathrm{nm}$ and $|C|=50\times 10^{-12}\mathrm{P}{\mathrm{a}}^{-1}$. Then $f_{\sigma }=12000N/m$. A3mm specimen at fringe order2.5 has stress-difference magnitude $2.5(12000)/0.003=10\mathrm{MPa}$. One additional complete phase cycle changes the inferred magnitude by4MPa for that thickness.

A uniaxial calibration supplies a known stress difference away from grips and other stress concentrations. For a rectangular strip of width10mm and thickness3mm, load300N gives stress10MPa. If the corresponding referenced fringe order is2.5, the measured fringe value is $10(3)/2.5=12\mathrm{MPa}\mathrm{mm}$. Repeating at several loads tests the assumed linear regime; counting from a referenced unloaded state helps separate the applied-load change from pre-existing retardation.

A primary instructional apparatus uses a manual leadscrew, load-cell readout, monochromatic illumination, polarizing elements and a camera. Those components connect a known mechanical load to the observed optical pattern.^[ref] A single intensity value does not uniquely identify an absolute phase cycle, so fringe order requires continuity, reference information or another phase-resolving method.

The magnitude convention loses the sign of $C$ and does not separately determine the two principal stresses. It also assumes the stated through-thickness optical/stress model. A varying stress field along the light path requires the appropriate retardation model rather than inserting one arbitrary local stress into the uniform-specimen formula.

Photoelasticity is the dependence of a material’s optical response on stress, used to reveal otherwise invisible mechanical patterns. A transparent material that is optically isotropic before loading can become birefringent under load: orthogonal polarizations acquire different refractive indices and therefore a relative phase retardation.^[ref]

For a thin, initially isotropic specimen in a suitable linear regime, the stress-optic law relates induced birefringence to the difference of in-plane principal stresses:Here $C$ is a material coefficient with units $\mathrm{P}{\mathrm{a}}^{-1}$, $t$ is the optical thickness, $\lambda$ the vacuum wavelength and $\delta$ the phase delay. This simple form assumes the relevant stress directions and difference do not change along the light path. A general three-dimensional specimen needs a more careful optical and mechanical model.

With ideal crossed polarizers, intensity normalized to the light after the first polarizer iswhere $\theta$ is the angle between the first polarization direction and a principal optical axis. A dark region can therefore arise from axis alignment or from an integer multiple of $2\pi$ retardation. Darkness alone does not prove that stress is zero.

For illustrative $C=3\times 10^{-12}\mathrm{P}{\mathrm{a}}^{-1}$, $t=5\mathrm{mm}$ and $\lambda =500\mathrm{nm}$, one full phase cycle corresponds toThese chosen values illustrate conversion from optical delay to a stress difference, not a calibration for every plastic. The method does not directly recover a hydrostatic stress shared equally by the two in-plane directions.

A polariscope makes the measurement visible. The Illinois teaching apparatus places a specimen between a polarizer and analyzer, applies load with a manual leadscrew, measures it with a load cell and records the pattern with a camera. Monochromatic illumination and adjustable polarization optics separate information about stress directions and differences.^[ref] Comparing patterns as load changes is a motivating experiment in connecting an optical observable to a mechanical model.

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]

The stress-optic law relates stress-induced birefringence to a mechanical stress difference. For an initially optically isotropic material in a suitable linear regime,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 $\mathrm{P}{\mathrm{a}}^{-1}$, and its wavelength dependence matters when comparing optical measurements.^[ref]

For uniform thickness $t$ and uniform relevant stress directions along the path, the measured phase retardation isThe 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 $s=d\delta /dF$, thenFor illustrative $s=0.003\mathrm{rad}{\mathrm{N}}^{-1}$, $\lambda =532\mathrm{nm}$, $A=20\mathrm{m}{\mathrm{m}}^2$ and $t=2\mathrm{mm}$, this gives $C\approx 2.54\times 10^{-12}\mathrm{P}{\mathrm{a}}^{-1}$. 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.^[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.