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 ${\varphi }_k$ 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.