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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.

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.