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Curvature bias in Gouy-phase measurements occurs when a spatially varying wavefront-curvature wave phase is mistaken for the axial Gouy phase in an interferometric comparison. For a fundamental Gaussian beam with time convention $e^{-i\omega t}$, the signal wave phase containsA curved reference contributes its own quadratic wave phase. After subtracting common axial path terms, the measured relative wave phase isThus a phase-retrieval algorithm can work exactly while the subsequent physical interpretation is biased. The Gaussian confinement wave phase and its arctangent dependence are derived in the primary theoretical treatment.^[ref]

For an original example with a planar reference and stable path wave phase, let $R_s(z)=z[1+(z_R/z{)}^2]$ and $z_R=k{w}_0^2/2$. At $z=z_R$ and fixed detector radius $r=w_0$, the curvature term is 0.5 radians. The true axial contribution is $-\pi /4$, whereas that off-axis pixel gives $-\pi /4+0.5\simeq -0.2854$ radians. At $z=-z_R$ the curvature term reverses. Comparing those two axial planes at this fixed radius gives a wave phase change $-\pi /2+1$, differing from the axial Gouy change by one radian.

The effect is observable even where the signal is appreciable: at $z=z_R$, $r=w_0$ is inside the plane's Gaussian radius $\sqrt{2}{w}_0$. Avoiding only dim pixels does not remove curvature bias.

A useful measurement fits the spatial quadratic wave phase, identifies the beam center and reference curvature, and extrapolates the relative wave phase to the axis. Tilt, decentering and path drift require their own terms; averaging several pixel phases is not automatically equivalent to measuring the on-axis wave phase. Phase unwrapping can be required before fitting a wave phase profile.

The primary 2023 Bessel-Gaussian experiment documents focusing optics and a rail-mounted camera in its interferometer.^[ref] It motivates tracking both propagation position and wavefront geometry, but its structured-mode observations must not be silently reclassified as measurements of the illustrative fundamental-Gaussian model above.

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.

Phase unwrapping reconstructs a wave phase field from measurements that determine only its value modulo $2\pi$. If a detector or complex-argument calculation returns ${\theta }_j=\mathrm{Arg}(e^{i{\varphi }_j})$, then all values ${\varphi }_j+2\pi n_j$ with integers $n_j$ give the same wrapped observation. Recovering a continuous wave phase therefore requires information beyond each isolated value. A primary 1982 analysis discusses this problem in wave phase curves obtained from image Fourier transforms before averaging them.^[ref]

Define $W(a)=\mathrm{Arg}(e^{ia})$ with principal interval $(-\pi ,\pi ]$. If adjacent true differences satisfy $|{\varphi }_{j+1}-{\varphi }_j|<\pi$, thenStarting from one reference wave phase, adding these wrapped differences reconstructs the sampled wave phase up to a global multiple of $2\pi$. The proof is direct: wrapping the two endpoint phases changes their difference only by an integer multiple of $2\pi$, and the strict bound selects its unique representative.

For an original example, true phases $0,0.8\pi ,1.6\pi ,2.4\pi$ wrap to $0,0.8\pi ,-0.4\pi ,0.4\pi$. Their wrapped increments are all $0.8\pi$, so reconstruction succeeds. But true increments $1.5\pi$ produce exactly the same wrapped samples as increments $-0.5\pi$. No algorithm can distinguish those two possibilities from the samples alone without a further assumption. Increasing sampling density can restore the bound only when the field's variation and noise allow it.

In two dimensions, different integration paths introduce a consistency test. A nonzero integer multiple of $2\pi$ in the sum of wrapped increments around a closed sampling loop is a phase-unwrapping residue. Noise, undersampling or true wave phase singularities can obstruct a single consistent smooth reconstruction. Choosing an integration path is therefore part of the inference.

For interferometry, low interference visibility can make the wave phase poorly determined even before unwrapping. Unwrapping does not recover information lost where the measured complex amplitude vanishes, nor does it distinguish an actual discontinuity from a smooth but insufficiently sampled change. Report the wave phase reference, principal interval, sampling and any continuity assumptions alongside the reconstructed field.