 Gouy phase

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.