Gouy phase

The Gouy phase is the phase shift that a focused beam accumulates relative to a plane wave of the same frequency traveling the same distance. For a Gaussian beam with Rayleigh range $z_R$ and its waist at $z = 0$, the on-axis phase is $kz - \zeta(z)$, with

$$\zeta(z) = \arctan\frac{z}{z_R}.$$

The Gouy phase runs from $-\pi/2$ far before the focus to $+\pi/2$ far after it, a total of $\pi$. It is zero at the beam waist, $\pi/4$ at $z = z_R$ and 0.46 rad (26.6°) at $z = z_R/2$. Half of the total is acquired within one Rayleigh range of the waist, and 80% within three. A beam focused in one dimension only, such as a line focus from a cylindrical lens, acquires $\pi/2$. The effect is named after Louis Georges Gouy, who observed it in 1890 by interfering a focused wave with an unfocused one.

Physical picture

The phase $kz - \zeta$ advances more slowly with $z$ than $kz$, so near the focus the on-axis wavefronts are spaced slightly more than one wavelength apart, equivalent to a phase velocity slightly above $c/n$. The local reduction of the axial wavevector is $d\zeta/dz$, which equals $1/z_R$ at the waist. For a 1064 nm beam with $z_R$ = 1 mm this is 1000 rad/m against $k = 5.9 \times 10^6$ rad/m, a relative change of $1.7 \times 10^{-4}$. A beam of finite width is a superposition of plane waves at small angles to the axis, each with an axial wavevector $k\cos\theta$ smaller than $k$, and the spread of angles grows as the beam narrows toward the focus; the shift is a property of diffraction and appears in focused waves of any kind.

Higher-order modes and cavity frequencies

Hermite-Gaussian modes TEM$_{mn}$ acquire $(m + n + 1)\,\zeta(z)$, and Laguerre-Gaussian modes $(2p + |l| + 1)\,\zeta(z)$. Higher-order transverse modes therefore slip in phase against the fundamental as they propagate. In a two-mirror resonator with stability parameters $g_i = 1 - L/R_i$, the one-way Gouy phase is $\arccos(\pm\sqrt{g_1 g_2})$, and the resonance condition gives

$$\begin{gathered} \nu_{qmn} = \frac{c}{2L}\left[\,q + (m + n + 1)\,\frac{\zeta_0}{\pi}\right],\\ \zeta_0 = \arccos\left(\pm\sqrt{g_1 g_2}\right), \end{gathered}$$

with the sign following that of $g_1$ and $g_2$. Adjacent transverse orders are separated by the fraction $\zeta_0/\pi$ of the longitudinal mode spacing $c/2L$. A 30 cm cavity with one flat mirror and one of 1 m radius has $g_1g_2 = 0.7$ and $\zeta_0/\pi$ = 0.185, so transverse orders are 92 MHz apart against a 500 MHz longitudinal spacing, as in the transverse-modes entry; the resonator stability entry gives the same spacing formula.

Confocal cavity. With $R_1 = R_2 = L$, both $g$ parameters are zero and the Gouy phase per pass is $\arccos 0 = \pi/2$. The transverse offset is then exactly half the longitudinal spacing: modes with even $m + n$ fall on the frequencies of TEM$_{00}$ and those with odd $m + n$ fall halfway between. The spectrum is a comb with spacing $c/4L$ whatever mixture of transverse modes is excited, which is why a scanning Fabry-Perot interferometer of confocal design tolerates imperfect mode matching. For $L$ = 5 cm, $c/2L$ = 3.0 GHz and the confocal comb has 1.5 GHz spacing. Other rational values also produce degeneracies: $g_1 g_2 = 0.25$ gives $\zeta_0/\pi = 1/3$, so every third transverse order coincides with a longitudinal mode.

Focused nonlinear optics

In harmonic generation with a focused beam, the driving polarization at the $q$-th harmonic carries $q\zeta$, while the generated harmonic beam carries roughly $\zeta$, so the Gouy phase adds a $z$-dependent term to the phase-matching condition. For third-harmonic generation in a uniform medium with normal dispersion and the focus well inside, the harmonic generated before the focus cancels that generated after it; an interface near the focus breaks the cancellation, which is the contrast mechanism of THG microscopy. In high-harmonic generation the Gouy term contributes to the wavevector mismatch, and the target position relative to the focus selects which electron trajectories phase-match. For few-cycle pulses, the Gouy phase shifts the carrier relative to the envelope by $\pi$ through the focus, so the carrier-envelope phase that a carrier-envelope offset lock stabilizes is defined only at a given position along the beam.

Measurement and pitfalls

The Gouy phase can be measured directly by interfering the focused beam with a plane or weakly focused reference and tracking the fringe position through the focus, where the pattern shifts by half a fringe. In a cavity it is read from the frequency difference between transverse modes, on a scanning interferometer or as a beat note on a fast photodiode, and this gives $g_1g_2$ for a cavity whose mirror curvatures are uncertain.

Sign conventions differ: texts that write the field as $e^{-i(kz - \omega t)}$ or $e^{i(kz - \omega t)}$ give the Gouy term opposite signs, and some write it as an excess phase delay, others as an advance. The magnitude and the total of $\pi$ are the same in all.

Common questions

Why is the Gouy phase shift π?

A focused beam changes from a converging to a diverging spherical wave, and the arctangent runs from $-\pi/2$ to $+\pi/2$ across that transition: the field emerges inverted relative to a plane wave. For a line focus the shift is $\pi/2$.

Does a collimated beam have a Gouy phase?

Yes, though it is acquired slowly: a beam with a 0.5 mm waist radius at 633 nm has $z_R$ = 1.24 m and gains $\pi/4$ over that distance.

References: L. G. Gouy, "Sur une propriété nouvelle des ondes lumineuses," C. R. Acad. Sci. Paris 110, 1251 (1890); A. E. Siegman, Lasers (University Science Books, 1986); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).