= Gouy phase interferometry
{c}
\b[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.<Gouy phase of Bessel-Gaussian beams: theory vs. experiment by Lyubomir Stoyanov and others>[Figure 3, printed p. 13691, apparatus caption]{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
$$
I_\delta=C+A\cos(\phi+\delta),\qquad
\boxed{\widehat\phi=\operatorname{atan2}(I_{3\pi/2}-I_{\pi/2},I_0-I_\pi).}
$$
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 $\phi=-\arctan\zeta$ after common path phases are removed. At $\zeta=1$, the four intensities are $(2.5,2.5,0.5,0.5)$, giving \b[$\widehat\phi=-\pi/4$]. At $\zeta=-1$, the first intensity is still $2.5$, but the quadratures reverse and the retrieved phase is $+\pi/4$.
\Image[/gouy-four-step-interferometry.png]
{title=Original synthetic four-step interferometry of a Gaussian beam: symmetric single-frame readings become distinguishable through phase quadratures. No experimental data are shown.}
<Curvature bias in Gouy-phase measurements> is important away from the axis: the signal adds $kr^2/(2R(z))$ 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>.
Back to article page
