Gouy phase shift

The Gouy phase shift is a phase shift gradually acquired by a Gaussian beam around its beam waist. It is named after Louis Georges Gouy.

Calculations

At distance from the waist of a Gaussian beam, the Gouy phase is given by
is the Rayleigh range:
where is the radius of the beam at the waist, is the index of refraction of the medium in which the beam propagates, and is the wavelength of the beam in free space.

The Gouy phase results in an increase in the apparent wavelength near the waist. Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light is small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature changes substantially over the interval of a single wavelength. In all cases the wave equation is satisfied at every position.
The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor. With dependence, the Gouy phase changes from to, while with dependence it changes from to along the axis.
For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to radians as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.