Gouy phase ($\psi (z)$)

The Gouy phase describes the additional axial phase associated with a focused wave relative to a plane wave. For a fundamental circular Gaussian beam with beam waist at $z=0$ and Rayleigh range $z_R>0$, defineWith the carrier convention $e^{i(kz-\omega t)}$, the on-axis field phase is $kz-\omega t-\psi (z)$ up to a constant. Thus the signed additional phase is $-\psi$ in this convention; defining the positive function $\psi$ does not reverse the physical prediction.^[ref]

The complex field prefactor gives a short derivation. The normalized Gaussian propagation factor isIts modulus describes the on-axis amplitude reduction; its argument is the additional phase. An intensity measurement alone loses this argument, so measuring the beam's width and brightness does not directly measure the Gouy phase.

As $z$ runs from far before to far after the waist, $\psi$ runs continuously from $-\pi /2$ to $+\pi /2$. The total additional phase change has magnitude $\pi$, but is not a discontinuous jump at the waist. For stipulated $w_0=100\mu \mathrm{m}$ and $\lambda =633\mathrm{nm}$, $z_R\approx 49.6\mathrm{mm}$. Between $z=-z_R$ and $z=+z_R$, the additional phase changes by $-\pi /2$; the far-field-to-far-field limit gives $-\pi$.

The number of confined transverse directions matters. A Gaussian beam with one transverse coordinate has prefactor $(1+iz/z_R{)}^{-1/2}$ and half this phase contribution. Simin Feng and Herbert G. Winful's primary analysis relates the effect to transverse confinement and the angular spectrum representation: a localized beam includes transverse wave-vector components, affecting axial propagation.^[ref]

An illustrative Gouy phase interferometry experiment overlaps the focused beam with a coherent reference beam and records the interference phase at different axial planes. Removing the ordinary optical-path contribution and accounting for wavefront curvature isolates the axial phase change. This is a proposed measurement outline, not an experimental result reported by the cited theoretical paper. In an optical cavity, related phase contributions help determine the resonance condition and distinguish transverse modes.

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.

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.