= Photoelastic branch-slip error
A <photoelastic branch-slip error> is an incorrect integer-cycle choice during <photoelastic phase unwrapping>. If the measured wrapped <phase retardation> is $\phi_k$ and the true unwrapped value is $\delta_k=\phi_k+2\pi m_k$, 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 $f_\sigma=10\,\mathrm{N\,mm^{-1}}$ 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.<Experimental Strain Analysis by David Roylance>[Printed pp.6–8, stress-optical law and crossed-polarizer fringe analysis]{ref}
A sufficient local noise margin is
$$
|\delta_{k+1}-\delta_k|+2\epsilon<\pi,
$$
when each phase error is bounded in magnitude by $\epsilon$. The triangle inequality bounds the error in the phase increment by $2\epsilon$, 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.<Polariscope and Photoelasticity by the University of Illinois Mechanical Testing Instructional Laboratory>[Apparatus description: manual loading, load cell and camera]{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.
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