7.8 Gouy phase

The equation for Hermite–Gauss modes shows an extra longitudinal phase lag. This Gouy phase [82625] 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
( z − z0 ) Ψ (z) = arctan ------ , (141 ) zR
or, using the Gaussian beam parameter,
( ) ℜ-{q}- Ψ (z ) = arctan ℑ {q} . (142 )
Compared to a plane wave, the phase lag φ of a Hermite–Gauss mode is
φ = (n + m + 1 )Ψ (z). (143 )
With an astigmatic beam, i.e., different beam parameters in the tangential and sagittal planes, this becomes
( ) ( ) 1 1 φ = n + 2- Ψt(z) + m + 2- Ψs(z), (144 )
with
( ) Ψ (z) = arctan ℜ-{qt} , (145 ) t ℑ {qt}
as the Gouy phase in the tangential plane (and Ψs is similarly defined in the sagittal plane).
View Image

Figure 45: These plots show the amplitude distribution and wave front (phase distribution) of helical Laguerre–Gauss modes upl. All plots refer to a beam with λ = 1 µm, w = 1 mm and distance to waist z = 1 m.

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