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# Gouy Phase Shift

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

German: Gouy-Phasenverschiebung

(Widespread wrong spelling: “Guoy phase shift”)

Along its propagation direction, a Gaussian beam acquires a phase shift which differs from that for a plane wave with the same optical frequency. This difference is called the Gouy phase shift [1]:

where zR is the Rayleigh length and z = 0 corresponds to the position of the beam waist. It results in a slightly increased distance between wavefronts, compared with the wavelength as defined for a plane wave of the same frequency. This also means that the phase fronts have to propagate somewhat faster, leading to an effectively increased local phase velocity.

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 π.

Figure 1: Beam radius and Gouy phase shift along the propagation direction for a beam in air with 1064 nm wavelength and 100 μm radius at the waist. 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.

For higher-order transverse modes, the Gouy phase shift is stronger. For TEMnm modes, for example, it is stronger by the factor 1 + n + m. This 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 aberrations [5].

### 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) [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) [5] R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions”, Opt. Express 14 (13), 6069 (2006) (explaining the important impact of the Gouy phase on the laser beam quality)

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