定义：
聚焦的高斯光束传播过程中产生的附加相移。 （常被错误的写为“Guoy Phase Shift”）

高斯光束在传播方向上会产生一个附加相移，这一相移与相同频率平面波产生的相移不同。这一不同的相移称为古依相移: 

其中 zR 为瑞利长度，而对应的是束腰所在的位置。它引起波前之间的距离稍微比同频率平面波波前之间距离大。这表明相位波前传播的更快，因此局域相速度更大。 
高斯光束通过焦点产生的总的古依相移（从一端的远场到焦点另一端的远场）为 π。 

图1：波长为1064nm，束腰半径为100μm的光束在空气中传播时，光束半径和古依相移在传播方向上的变化。位置的正负是相对于瑞利长度而言的。 
高斯光束的相移与平面波相移不同是很正常的。高斯光束可看做不同方向传播的平面波的叠加。不在光轴方向传播的平面波成分的相移要比光轴方向的平面波相移小，而总体的相移来自于所有叠加的平面波成分。 

高阶横模的古依相移更大。例如，对于TEMnm模，其增强因子为 1 + n + m。因此光学谐振腔中的高阶模式的共振频率更高。由于会使谐振腔模式的频率简并解除，古依相移会影响激光器谐振腔的光束质量.

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

More general terms: optical phase shift

(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 z_R is the Rayleigh length and z = 0 corresponds to the position of the beam waist. 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. This also means 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 −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.

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 (TEM_nm) 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].

Bibliography