\b[<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> contains
$$
\phi_s(r,z)=kz+\frac{kr^2}{2R_s(z)}-\psi(z),\qquad
\psi(z)=\arctan(z/z_R).
$$
A curved reference contributes its own quadratic <wave phase>. After subtracting common axial path terms, the measured relative <wave phase> is
$$
\phi_{\rm rel}(r,z)=-\psi(z)+\frac{kr^2}{2}\left[\frac1{R_s(z)}-\frac1{R_r(z)}\right]+\phi_{\rm path}(z).
$$
Thus 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.<Physical origin of the Gouy phase shift by Simin Feng and Herbert G. Winful>[Equations 4–12, Gaussian field phase and confinement]{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 $z_R=kw_0^2/2$. 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> $\sqrt2w_0$. Avoiding only dim pixels does not remove curvature bias.

\Image[/gouy-phase-curvature-bias.png]
{title=Original Gaussian-beam model showing off-axis curvature bias in a Gouy-phase comparison}
{description=The reference wavefront is planar and detector radius is fixed relative to the waist radius. At radius equal to the waist radius, curvature contributes plus or minus 0.5 radians at one Rayleigh range from focus. The curves are calculated examples, not experimental data.}

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.<Gouy phase of Bessel-Gaussian beams: theory vs. experiment by Lyubomir Stoyanov and others>[Figure 3, p. 13691]{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.


 Curvature bias in Gouy-phase measurements (source code)