Editing Gaussian beam (section)

== Beam parameters ==

The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength {{mvar|λ}} (''in'' the dielectric medium, if not free space) and the following '''beam parameters''', all of which are connected as detailed in the following sections.

===Beam waist===<!--Beam waist redirects here-->
{{see also|Beam diameter}}
[[Image:GaussianBeamWaist.svg|thumb|upright=1.5|right|Gaussian beam width {{math|''w''(''z'')}} as a function of the distance {{mvar|z}} along the beam, which forms a [[hyperbola]]. {{math|''w''<sub>0</sub>}}: beam waist; {{mvar|b}}: depth of focus; {{math|''z''<sub>R</sub>}}: [[Rayleigh range]]; {{mvar|Θ}}: total angular spread]]

The shape of a Gaussian beam of a given wavelength {{mvar|λ}} is governed solely by one parameter, the ''beam waist'' {{math|''w''<sub>0</sub>}}. This is a measure of the beam size at the point of its focus ({{math|1=''z'' = 0}} in the above equations) where the beam width {{math|''w''(''z'')}} (as defined above) is the smallest (and likewise where the intensity on-axis ({{math|1=''r'' = 0}}) is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the [[Rayleigh range]] {{math|''z''<sub>R</sub>}} and asymptotic beam divergence {{mvar|θ}}, as detailed below.

===Rayleigh range and confocal parameter===
{{main|Rayleigh length}}
The ''Rayleigh distance'' or ''Rayleigh range'' {{math|''z''<sub>R</sub>}} is determined given a Gaussian beam's waist size:

<math display="block">z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}.</math>

Here {{mvar|λ}} is the wavelength of the light, {{mvar|n}} is the index of refraction. At a distance from the waist equal to the Rayleigh range {{math|''z''<sub>R</sub>}}, the width {{mvar|w}} of the beam is {{math|{{sqrt|2}}}} larger than it is at the focus where {{math|1=''w'' = ''w''<sub>0</sub>}}, the beam waist. That also implies that the on-axis ({{math|1=''r'' = 0}}) intensity there is one half of the peak intensity (at {{math|1=''z'' = 0}}). That point along the beam also happens to be where the wavefront curvature ({{math|1/''R''}}) is greatest.<ref name="svelto153" />

The distance between the two points {{math|1=''z'' = ±''z''<sub>R</sub>}} is called the ''confocal parameter'' or ''depth of focus'' of the beam.<ref>{{cite journal | last=Brorson | first=S.D. | date=1988 | title=What is the confocal parameter? | url=https://ieeexplore.ieee.org/document/155|journal=IEEE Journal of Quantum Electronics | volume=24 | issue=3 | pages=512–515 | doi=10.1109/3.155| bibcode=1988IJQE...24..512B }}</ref>

===Beam divergence===
{{Further|Beam divergence}}
Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where {{math|1=''r'' = ''w''(''z'')}}. That is where the intensity has dropped to {{math|1/''e''<sup>2</sup>}} of its on-axis value. Now, for {{math|''z'' ≫ ''z''<sub>R</sub>}} the parameter {{math|''w''(''z'')}} increases linearly with {{mvar|z}}. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose {{math|1=''r'' = ''w''(''z'')}}) and the beam axis ({{math|''r'' {{=}} 0}}) defines the ''divergence'' of the beam:
<math display="block">\theta = \lim_{z\to\infty} \arctan\left(\frac{w(z)}{z}\right).</math>

In the paraxial case, as we have been considering, {{mvar|θ}} (in radians) is then approximately<ref name="svelto153" />
<math display="block">\theta = \frac{\lambda}{\pi n w_0}</math>

where {{mvar|n}} is the refractive index of the medium the beam propagates through, and {{mvar|λ}} is the free-space wavelength. The total angular spread of the diverging beam, or ''apex angle'' of the above-described cone, is then given by
<math display="block">\Theta = 2 \theta\, .</math>

That cone then contains 86% of the Gaussian beam's total power.

Because the divergence is inversely proportional to the spot size, for a given wavelength {{mvar|λ}}, a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to ''minimize'' the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section ({{math|''w''<sub>0</sub>}}) at the waist (and thus a large diameter where it is launched, since {{math|''w''(''z'')}} is never less than {{math|''w''<sub>0</sub>}}). This relationship between beam width and divergence is a fundamental characteristic of [[diffraction]], and of the [[Fourier transform]] which describes [[Fraunhofer diffraction]]. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.

Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam.<ref>Siegman (1986) p. 630.</ref> From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about {{math|2''λ''/''π''}}.

[[Laser beam quality]] is quantified by the [[beam parameter product]] (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size {{math|''w''<sub>0</sub>}}. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as {{math|''M''<sup>2</sup>}} ("[[M squared]]"). The {{math|''M''<sup>2</sup>}} for a Gaussian beam is one. All real laser beams have {{math|''M''<sup>2</sup>}} values greater than one, although very high quality beams can have values very close to one.

The [[numerical aperture#Laser physics|numerical aperture]] of a Gaussian beam is defined to be {{math|1=NA = ''n'' sin ''θ''}}, where {{mvar|n}} is the [[index of refraction]] of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by 
<math display="block">z_\mathrm{R} = \frac{n w_0}{\mathrm{NA}} .</math>

===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"/>