 Photoelastic phase unwrapping

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.