Phase-shifting interferometry

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 iswhere $C$ 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 and compensation of instrumental wavefront errors.^[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 asturns wave phase recovery into a linear fit followed by $\varphi =atan2(-D,B)$. 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.

Four-step phase retrieval estimates relative wave phase from four wave interference frames separated by quarter-cycle wave phase steps. Use the explicit signal-arm conventionwith a common background $C$ and modulation amplitude $A>0$. Opposite-frame differences giveso $\hat{\varphi }=atan2(Y,X)$. The order of the subtractions fixes the sign. Changing which arm receives the wave phase step requires a correspondingly changed convention.

The same data estimate $C=(I_0+I_{\pi /2}+I_{\pi }+I_{3\pi /2})/4$ and $A=\sqrt{X^2+Y^2}/2$. A simple model-consistency residual is $(I_0+I_{\pi })-(I_{\pi /2}+I_{3\pi /2})$, which is zero for the exact stated model. It does not detect every possible systematic error.

For an original example, let $C=3$, $A=2$ and $\varphi =\pi /3$. The four light intensities are $4$, $3-\sqrt{3}$, $2$ and $3+\sqrt{3}$. Then $X=2$, $Y=2\sqrt{3}$, and the recovered wave phase is $\pi /3$. Common background cancels, whereas changing background separately between frames need not cancel.

If independent additive errors in the four light intensities have common variance ${\sigma }_I^2$, first-order propagation givesThis follows from $d\varphi =(XdY-YdX)/(X^2+Y^2)$ and difference variances $2{\sigma }_I^2$. It applies away from vanishing modulation and assumes the specified independent equal-noise model. wave phase uncertainty grows when interference visibility is weak.

The method is a discrete instance of phase-shifting interferometry, whose early digital apparatus used controlled reference-path modulation and synchronous wave phase extraction.^[ref] The result is still a principal wave phase; phase unwrapping cannot be replaced by choosing a different arctangent branch at each pixel without a consistency rule.