= Phase-shifting interferometry
\b[<Phase-shifting interferometry>] measures an optical <phase difference> by recording <light intensities> at controlled relative <wave phase> shifts. For two sufficiently coherent fields at one detector point, a common model is
$$
I_j=C+A\cos(\phi+\delta_j),\qquad A\ge0,
$$
where $C$ is a common background, $A$ a modulation amplitude and $\delta_j$ a known applied step. Several measurements separate the unknown <wave phase> from <light intensity> background and contrast; a single bright or dark frame generally cannot do so.
A concrete primary implementation was reported in 1974. A piezoelectrically varied reference arm and a 1024-element photodiode array supplied sampled <light intensities> to a minicomputer, which recovered <wave phase> by synchronous Fourier-series detection. The paper describes optical-surface and lens testing, repeated <sampling (signal processing)> and compensation of instrumental <wavefront> errors.<Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses by J. H. Bruning and others>[Abstract and p. 2693, apparatus and phase-detection method]{ref} This is an experimental implementation of controlled <wave phase> modulation; it is not a claim that every such system uses the same number of steps.
Writing the model as
$$
I_j=C+B\cos\delta_j+D\sin\delta_j,\qquad
B=A\cos\phi,\quad D=-A\sin\phi,
$$
turns <wave phase> recovery into a linear fit followed by $\phi=\operatorname{atan2}(-D,B)$. The step choices must make the three columns $1,\cos\delta_j,\sin\delta_j$ independent. Repeating only the same <wave phase> cannot identify all three unknowns. With more steps, <least squares> can fit the coefficients while exposing model residuals.
A reflecting reference mirror moved axially by $\Delta z$ changes its round-trip optical path by $2\Delta z$ in the simple normal-incidence geometry, giving $\delta=4\pi\Delta z/\lambda$. A quarter-cycle <wave phase> step thus corresponds to $\Delta z=\lambda/8$; at 632.8 nm this is 79.1 nm. These are geometric model values, not a calibration result for the historical apparatus.
<Phase-step calibration error>, <light intensity> drift, motion between frames and detector nonlinearity can all break the common-intensity model. The retrieved <wave phase> remains modulo $2\pi$, so <phase unwrapping> requires additional spatial or temporal information. The useful observable is the relative <wave phase> under controlled acquisition conditions, not an absolute optical <wave phase> inferred from brightness alone.
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