### 7.8 Gouy phase

The equation for Hermite–Gauss modes shows an extra longitudinal phase lag. This Gouy
phase [8, 26, 25] describes the fact that, compared to a plane wave, the Hermite–Gauss modes have a
slightly slower phase velocity, especially close to the waist. The Gouy phase can be written as
or, using the Gaussian beam parameter,
Compared to a plane wave, the phase lag of a Hermite–Gauss mode is
With an astigmatic beam, i.e., different beam parameters in the tangential and sagittal planes, this becomes
with
as the Gouy phase in the tangential plane (and is similarly defined in the sagittal plane).