Photoelastic fringe order ($N=|\delta |/(2\pi )$)

The photoelastic fringe order $N$ expresses the magnitude of the full relative phase retardation between two principal optical components in cycles: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.

Photoelastic phase unwrapping assigns consistent cycle counts to retardation observations that determine phase only modulo a full cycle. If a phase-resolved method reports ${\varphi }_k\in (-\pi ,\pi ]$, the full relative phase retardation has the formThe integers carry the fringe-order information that the wrapped values omit. A known starting order and sufficiently small changes can make the branch choices recoverable.

For an original one-dimensional example, take wrapped phases $2.80,3.05,-2.95,-2.70$ radians. Choosing the first value as the known starting branch, the nearby increasing continuation isor approximately $2.80,3.05,3.3332,3.5832$ radians. The apparent large negative jump is a change of representation at $\pi$, not a demonstrated physical reversal. The final photoelastic fringe order on this branch is about0.5703.

A nearest-branch rule is justified in a noiseless sequence when successive true phase changes have magnitude strictly below $\pi$: among increments differing by $2\pi$, exactly one lies in that interval. With phase errors bounded by $ϵ$, the stronger condition $|{\delta }_{k+1}-{\delta }_k|+2ϵ<\pi$ is sufficient for that local choice. Larger steps, discontinuities or inadequate signal can cause a photoelastic branch-slip error. One wrong cycle changes inferred stress-difference magnitude by $f_{\sigma }/t$ in the uniform-specimen model.

The algorithm assumes phase or equivalent branch information is available. A single crossed-polarizer intensity $I/I_0={\sin }^2(\pi N)$ does not itself supply a unique signed wrapped phase. Its ambiguity includes reflection within a cycle as well as the integer count. Additional polarization measurements, a known load path or other reference information must justify phase recovery before applying the simple continuation rule. The primary stress-analysis derivation explicitly shows how intensity depends on retardation and axis orientation.^[ref]

The Illinois apparatus supplies controlled loading, a load-cell record and camera observations, making a load-indexed sequence physically meaningful.^[ref] Continuity alone still leaves a common $2\pi$ offset unless the absolute starting order is known. An unloaded image can reveal residual retardation rather than automatically establish zero stress or zero order.

A photoelastic branch-slip error is an incorrect integer-cycle choice during photoelastic phase unwrapping. If the measured wrapped phase retardation is ${\varphi }_k$ and the true unwrapped value is ${\delta }_k={\varphi }_k+2\pi m_k$, replacing the correct integer $m_k$ by $m_k+j$ changes the recovered phase by $2\pi j$. A small error in one branch decision can therefore become a full-cycle offset in many later observations.

For an original noiseless counterexample, suppose a known starting phase is $2.2\pi$ and the next true phase is $3.4\pi$. Their wrapped values in $(-\pi ,\pi ]$ are $0.2\pi$ and $-0.6\pi$. The nearest-increment rule chooses $-0.8\pi$ instead of the true $1.2\pi$, producing $1.4\pi$ as the second recovered phase. The recovered photoelastic fringe order is $0.7$ rather than $1.7$. Every later small increment can be locally consistent while retaining that wrong cycle.

In the uniform-thickness stress-optical model, $N=t|{\sigma }_1-{\sigma }_2|/f_{\sigma }$, where $f_{\sigma }$ is the material fringe value. While staying on a known nonnegative stress-difference branch, one cycle changes the inferred stress difference by $f_{\sigma }/t$. Stipulated $f_{\sigma }=10\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$ and thickness $t=5\mathrm{mm}$ make the example's one-cycle error $2\mathrm{MPa}$. The material coefficient depends on the optical conditions, including wavelength; the primary photoelastic derivation explains that dependence and the analyzer's periodic intensity response.^[ref]

A sufficient local noise margin iswhen each phase error is bounded in magnitude by $ϵ$. The triangle inequality bounds the error in the phase increment by $2ϵ$, leaving it inside the nearest-branch interval. Equality gives no strict guarantee. Reducing the loading increment helps only if the assumed phase recovery and error bound are valid.

The Illinois polariscope provides controlled loading, a load readout and camera observation, which support a proposed check using smaller repeated load increments.^[ref] Agreement between a coarse sequence and a finer sequence tests branch stability over those measurements. It cannot establish an unknown common starting order. A lone crossed-polarizer intensity also does not supply a unique signed phase: the additional phase-recovery information assumed by the algorithm must be measured or independently justified.