 Curvature bias in Gouy-phase measurements

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.