= 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 $\phi_k\in(-\pi,\pi]$, the full relative <phase retardation> has the form
$$
\delta_k=\phi_k+2\pi m_k,\qquad m_k\in\mathbb Z.
$$
The 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 is
$$
2.80,\ 3.05,\ 2\pi-2.95,\ 2\pi-2.70,
$$
or 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 $\epsilon$, the stronger condition $|\delta_{k+1}-\delta_k|+2\epsilon<\pi$ 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.<Experimental Strain Analysis by David Roylance>[Printed pp.6–8, analyzer transmission and fringe interpretation]{ref}
The Illinois apparatus supplies controlled loading, a load-cell record and camera observations, making a load-indexed sequence physically meaningful.<Polariscope and Photoelasticity by the University of Illinois Mechanical Testing Instructional Laboratory>[Manual leadscrew/load readout and camera observation]{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.
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