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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.

Gouy phase interferometry compares a focused field with a coherent reference to recover the additional Gouy phase. Beam brightness alone discards the phase. An interferometer converts a relative field phase into intensity modulation, but the optical paths and reference wavefront curvature must be modeled.

A published 2023 experiment by Lyubomir Stoyanov and colleagues used a Mach-Zehnder interferometer to study Bessel-Gaussian beams. Its setup included a 532-nanometre laser, beam expansion, two spatial light modulators, focusing lenses and a CCD camera on a rail to follow propagation after the focus. Its Figure 3 also describes a single-lens interferometer arrangement for stronger focusing.^[ref] Structured modes need their own phase model; that experiment should not be silently interpreted as a fundamental-Gaussian measurement.

A separate original protocol illustrates four-step phase retrieval. At one detector point, apply controlled signal-arm phases $\delta =0,\pi /2,\pi ,3\pi /2$ while the beam intensities remain fixed. Write
This phase-shifting interferometry formula removes the unknown common intensity background $C$ and positive modulation amplitude $A$. It returns phase modulo $2\pi$; phase unwrapping requires continuity information between sampled positions. If the interference visibility vanishes, the phase is not observable by this formula.

For a synthetic coherent Gaussian/reference pair, use reference intensity 1 and signal intensity $1/(1+{\zeta }^2)$, with $\zeta =z/z_R$. Then $C=1+1/(1+{\zeta }^2)$, $A=2/\sqrt{1+{\zeta }^2}$ and $\varphi =-\arctan \zeta$ after common path phases are removed. At $\zeta =1$, the four intensities are $(2.5,2.5,0.5,0.5)$, giving $\hat{\varphi }=-\pi /4$. At $\zeta =-1$, the first intensity is still $2.5$, but the quadratures reverse and the retrieved phase is $+\pi /4$.

Curvature bias in Gouy-phase measurements is important away from the axis: the signal adds $k{r}^2/(2R(z))$ to its spatial phase. Fit or remove that term when using multiple pixels, retain calibrated phase steps and account for drift between frames. The recovered phase combines all coherent optical-path contributions until those controls isolate the Gouy phase.

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.