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# Gouy Phase Shift

Author: the photonics expert

Definition: an additional phase shift occurring in the propagation of focused Gaussian beams

More general term: optical phase shift

(Widespread wrong spelling: “Guoy phase shift”)

A plane wave propagating in <$z$> direction acquires an phase shift <$k z$> according to its wavenumber <$k$>. This rule does not precisely hold for light beams, being confined in the transverse directions: Here, the phase evolution somewhat deviates from that of a plane wave, and the phase advancement per unit length is not constant, particularly around the focus.

In particular, for a Gaussian beam the phase shift deviates from that of a plane wave by the so-called Gouy phase shift [1]:

$${\varphi _{\rm{G}}}(z) = - \arctan \frac{z}{{{z_{\rm{R}}}}}$$

where <$z_\textrm{R}$> is the Rayleigh length and <$z$> = 0 corresponds to the position of the beam waist. As can be seen in Figure 1, that term varies substantially around the beam focus, and not much in regions far from the focus.

In the literature, the formula is also found with a positive sign, but in any case the phase shift of a Gaussian beam is reduced (rather than increased) with respect to that of a plane wave. This results in a slightly increased distance between wavefronts, compared with the wavelength as defined for a plane wave of the same frequency. A consequence is that the phase fronts have to propagate somewhat faster, leading to an effectively increased local phase velocity. For propagation in vacuum, that amounts to a kind of superluminal transmission.

Overall, the Gouy phase shift of a Gaussian beam for going through a focus (from the far field to the far field on the other side of the focus) is <$-\pi$>, as that phase shift changes from the asymptotic value <$+\pi / 2$> to <$-\pi / 2$>.

The positions at plus and minus the Rayleigh length are marked.

It is actually not surprising that the phase shift of a Gaussian beam is not exactly the same as for a plane wave. A Gaussian beam can be considered as a superposition of plane waves with different propagation directions. Those plane wave components with propagation directions different from the beam axis experience smaller phase shifts in <$z$> direction; the overall phase shift arises from a superposition of all these components.

A similar kind of reasoning is based on the observation that the transverse confinement causes some spread of transverse wave vector components, thus a corresponding reduction in the longitudinal wave vector component, which finally leads to a reduction of the phase shift [5].

For higher-order transverse modes, there is a similar but stronger phase shift. For Hermite–Gaussian (TEMnm) modes, for example, it is stronger by the factor <$1 + n + m$>, while for Laguerre–Gaussian modes that factor is <$1 + |m| + 2 p$>, where <$p$> is the radial index and <$m$> the azimuthal index. This change causes the resonance frequencies of higher-order modes in optical resonators to be somewhat higher. By lifting the frequency degeneracy of resonator modes, the Gouy phase shift also has an important impact on the beam quality achieved in a laser resonator under the influence of optical aberrations [6].

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### Bibliography

 [1] L. G. Gouy, “Sur une propriete nouvelle des ondes lumineuses”, C. R. Acad. Sci. Paris 110, 1251 (1890) [2] A. Rubinowicz, “On the anomalous propagation of phase in the focus”, Phys. Rev. 54 (11), 931 (1938); https://doi.org/10.1103/PhysRev.54.931 [3] L. G. Gouy, “Sur la propagation anomale des ondes”, Compt. Rendue Acad. Sci. Paris 111, 33 (1890) [4] R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams”, J. Opt. Soc. Am. 70 (7), 877 (1980); https://doi.org/10.1364/JOSA.70.000877 [5] S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift”, Opt. Lett. 26 (8), 485 (2001); https://doi.org/10.1364/OL.26.000485 [6] R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions”, Opt. Express 14 (13), 6069 (2006); https://doi.org/10.1364/OE.14.006069 (explaining the important impact of the Gouy phase on the laser beam quality) [7] M. Hiekkamäki et al., “Observation of the quantum Gouy phase”, Nature Photonics 16, 828–833 (2022); https://doi.org/10.1038/s41566-022-01077-w