# Gouy Phase Shift

The first two terms in the general expression for the Gaussian beam are:

\begin{equation} e^{-ikz-i\phi(z)} \end{equation}

where $tan\phi(z) = z/z_R$. The term $\phi(z)$ describes a phase shift that the Gaussian envelope acquires through axial propagation if transversely confined. This is called the Gouy phase shift, and it means that a confined beam experiences a negative phase shift on propagation. For a beam propagating from its waist to the far field, the phase shift is $\pi/2$; for a beam propagating through focus, the phase shift is $\pi$. Half of the phase shift occurs in the Rayleigh range of the beam waist, which makes sense because the phase shift arises from tight confinement and the beam is most tightly confined in the Rayleigh range.

First, we can think quantum mechanically about what it would mean for a beam to acquire a phase shift once confined, as the main two parameters at play here are position and momentum which must follow uncertainty principles. For an infinite plane wave, there are no transverse momentum components as the propagation is purely in the z-direction. Thus, the transverse position has infinite extent by the uncertainty principle. For a Gaussian beam, however, there are transverse momentum components, which means that the position would not be infinite in extent. The Gouy phase shift can then be seen as the expectation value of the axial position shift due to the transverse momentum components.

Another interpretation would be through the angular spectrum picture. Because a Gaussian beam can be thought of as a superposition of plane wave components, the plane wave components that do not lie directly along the z-axis would experience a smaller phase shift in comparison to a pure plane wave, and thus the superposition of each of the plane wave components results in a decreased axial phase shift. Interestingly, a decreased axial phase shift means that the wavefronts have an increased distance between them, meaning effectively that the phase front has to travel faster locally.

The best paper on this subject is S. Feng and H. Winful, Opt. Lett. 26, 485 (2001). This paper gives the phase shift for any arbitrary beam. It should be noted that the phase shift occurs for any beam that is tightly confined in one or more dimensions, and is not limited to optical beams.

If we consider the case where we have a cylindrical lens focusing just the x-axis of a Gaussian beam, then we would only have confinement in the x-direction, meaning then that we would only have a $\pi/4$ shift from focus to the far field.

When is the Gouy phase shift important? The axial mode frequencies of optical resonators are formed by spherical mirrors and are directly affected by the Gouy shift.