= Gouy phase
{c}
{title2=$\psi(z)$}
= Gouy phase shift
{c}
{synonym}
The \b[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> $z_R>0$, define
$$
\psi(z)=\arctan\!\left(\frac{z}{z_R}\right),\qquad
z_R=\frac{\pi w_0^2}{\lambda}.
$$
With 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.<Physical origin of the Gouy phase shift by Simin Feng and Herbert G. Winful>[Equation12, printed p.486, sign convention for the Gaussian phase contribution]{ref}
The complex field prefactor gives a short derivation. The normalized Gaussian propagation factor is
$$
\frac1{1+iz/z_R}
=\frac{1}{\sqrt{1+(z/z_R)^2}}\,e^{-i\arctan(z/z_R)}.
$$
Its 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 $w_0=100\,\mathrm{\mu m}$ and $\lambda=633\,\mathrm{nm}$, $z_R\approx49.6\,\mathrm{mm}$. 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.<Physical origin of the Gouy phase shift by Simin Feng and Herbert G. Winful>[Equations4–12 and line-focus versus point-focus discussion, pp.485–486]{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.
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