Editing Gaussian beam (section)

===Gouy phase===
The ''[[Louis Georges Gouy|Gouy]] phase'' is a phase shift gradually acquired by a beam around the focal region. At position {{mvar|z}} the Gouy phase of a fundamental Gaussian beam is given by<ref name="svelto153" />
<math display="block">\psi(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right).</math>

[[File:Bildschirmfoto 2020-07-05 um 12.50.52.png|thumb|Gouy phase.]]
The Gouy phase results in an increase in the apparent wavelength near the waist ({{math|''z'' ≈ 0}}).  Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a [[Near and far field|near-field]] phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large [[numerical aperture]], in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the [[Helmholtz equation|wave equation]] is satisfied at every position.

The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor.<ref name="gouy_phase_shift" /> With  {{math|''e''<sup>''iωt''</sup>}} dependence, the Gouy phase changes from {{math|-''π''/2}} to {{math|+''π''/2}}, while with {{math|''e''<sup>-''iωt''</sup>}} dependence it changes from {{math|+''π''/2}} to {{math|-''π''/2}} 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 {{mvar|π}} radians (thus a phase reversal) 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 [[#Laguerre-Gaussian modes|higher-order Gaussian modes]].<ref name="gouy_phase_shift"/>