Editing Gaussian beam (section)

==Mathematical form==

The equations below assume a beam with a circular cross-section at all values of {{mvar|z}}; this can be seen by noting that a single transverse dimension, {{mvar|r}}, appears.  Beams with [[Elliptic curve|elliptical]] cross-sections, or with waists at different positions in {{mvar|z}} for the two transverse dimensions ([[Astigmatism (optical systems)|astigmatic]] beams) can also be described as Gaussian beams, but with distinct values of {{math|''w''<sub>0</sub>}} and of the {{math|1 = ''z'' = 0}} location for the two transverse dimensions {{mvar|x}} and {{mvar|y}}.

[[Image:Gaussian-beam intensity surfaceplot.png|thumb|Gaussian beam intensity profile with {{math|''w''<sub>0</sub> {{=}} 2''λ''}}.]]
The Gaussian beam is a [[transverse electromagnetic mode|transverse electromagnetic (TEM) mode]].<ref name="svelto158">Svelto, p. 158.</ref> The mathematical expression for the electric field amplitude is a solution to the [[Helmholtz equation#Paraxial approximation|paraxial Helmholtz equation]].<ref name="svelto153" /> Assuming polarization in the {{mvar|x}} direction and propagation in the {{math|+''z''}} direction, the electric field in [[phasor]] (complex) notation is given by:

<math display="block">{\mathbf E(r,z)} = E_0 \, \hat{\mathbf x} \, \frac{w_0}{w(z)}
\exp \left( \frac{-r^2}{w(z)^2}\right )
\exp \left(\! -i \left(kz +k \frac{r^2}{2R(z)} - \psi(z) \right) \!\right)</math>

where<ref name="svelto153" /><ref>{{Cite book|last1=Yariv |first1=Amnon |first2=Albert Pochi |last2=Yeh |title=Optical Waves in Crystals: Propagation and Control of Laser Radiation |date=2003 |publisher=J. Wiley & Sons |isbn=0-471-43081-1 |oclc=492184223}}</ref>

*{{mvar|r}} is the radial distance from the center axis of the beam,
*{{mvar|z}} is the axial distance from the beam's focus (or "waist"),
*{{mvar|i}} is the [[imaginary unit]],
*{{math|1=''k'' = 2''πn''/''λ''}} is the [[wave number]] (in [[radian]]s per meter) for a free-space wavelength {{mvar|λ}}, and {{mvar|n}} is the index of refraction of the medium in which the beam propagates,
*{{math|1=''E''<sub>0</sub> = ''E''(0, 0)}}, the electric field amplitude at the origin ({{math|1=''r'' = 0}}, {{math|1=''z'' = 0}}),
*{{math|''w''(''z'')}} is the radius at which the field amplitudes fall to {{math|1/''e''}} of their axial values (i.e., where the intensity values fall to {{math|1/''e''<sup>2</sup>}} of their axial values), at the plane {{mvar|z}} along the beam,
*{{math|1=''w''<sub>0</sub> = ''w''(0)}} is the [[#Beam waist|waist radius]],
*{{math|''R''(''z'')}} is the [[#Wavefront curvature|radius of curvature]] of the beam's [[wavefront]]s at {{mvar|z}}, and
*{{math|1=''ψ''(''z'') = arctan(''z''/''z''<sub>R</sub>)}} is the [[#Gouy phase|Gouy phase]] at {{mvar|z}}, an extra phase term beyond that attributable to the [[phase velocity]] of light.

The physical electric field is obtained from the phasor field amplitude given above by taking the real part of the amplitude times a time factor:
<math display=block>\mathbf E_\text{phys}(r,z,t) = \operatorname{Re}(\mathbf E(r,z) \cdot e^{i\omega t}),</math>
where <math display=inline>\omega</math> is the [[angular frequency]] of the light and {{mvar|t}} is time. The time factor involves an arbitrary [[sign convention]], as discussed at {{section link|Mathematical descriptions of opacity|Complex conjugate ambiguity}}.

Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where {{math|''w''<sub>0</sub> ≫ ''λ''/''n''}}.

The corresponding [[intensity (physics)|intensity]] (or [[irradiance]]) distribution is given by

<math display="block"> I(r,z) = { |E(r,z)|^2  \over  2 \eta } = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w(z)^2}\right),</math>

where the constant {{mvar|η}} is the [[wave impedance]] of the medium in which the beam is propagating. For free space, {{math|1=''η'' = [[Impedance of free space|''η''<sub>0</sub>]]}} ≈ 377 Ω. {{math|1=''I''<sub>0</sub> = {{mabs|''E''<sub>0</sub>}}<sup>2</sup>/2''η''}} is the intensity at the center of the beam at its waist.

If {{math|''P''<sub>0</sub>}} is the total [[Power (physics)|power]] of the beam,
<math display="block">I_0 = {2P_0 \over \pi w_0^2}.</math>

===Evolving beam width===
[[File:Gaussian Beam FWHM.gif|thumb|upright=1.5|The [[Gaussian function]] has a {{math|1/''e''<sup>2</sup>}} diameter ({{math|2''w''}} as used in the text) about 1.7 times the [[FWHM]].]]
At a position {{mvar|z}} along the beam (measured from the focus), the spot size parameter {{mvar|w}} is given by a [[Hyperbola#Equation|hyperbolic relation]]:<ref name="svelto153" />
<math display="block">w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 },</math>
where<ref name="svelto153" />
<math display="block">z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}</math>
is called the [[Rayleigh range]] as further discussed below, and <math>n</math> is the refractive index of the medium.

The radius of the beam {{math|''w''(''z'')}}, at any position {{mvar|z}} along the beam, is related to the [[full width at half maximum]] (FWHM) of the intensity distribution at that position according to:<ref name=zemax>{{cite web |url=http://www.zemax.com/support/resource-center/knowledgebase/how-to-convert-fwhm-measurements-to-1-e-squared-ha |title=How to Convert FWHM Measurements to 1/e-Squared Halfwidths |first=Dan |last=Hill |date=April 4, 2007 |work=Radiant Zemax Knowledge Base |access-date=June 7, 2016 |archive-date=March 4, 2016 |archive-url=https://web.archive.org/web/20160304035034/http://www.zemax.com/support/resource-center/knowledgebase/how-to-convert-fwhm-measurements-to-1-e-squared-ha |url-status=dead }}</ref>
<math display="block">w(z)={\frac {\text{FWHM}(z)}{\sqrt {2\ln2}}}.</math>

===Wavefront curvature===
The wavefronts have zero curvature (radius = ∞) at the waist. Wavefront curvature increases away from the waist, with the maximum rate of change occurring at the Rayleigh distance, {{math|1=''z'' = ±''z''<sub>R</sub>}}. Beyond the Rayleigh distance, {{math|{{mabs|''z''}} > ''z''<sub>R</sub>}}, the curvature again decreases in magnitude, approaching zero as {{math|''z'' → ±∞}}. The curvature is often expressed in terms of its reciprocal, {{mvar|R}}, the ''[[Radius of curvature (optics)|radius of curvature]]''; for a fundamental Gaussian beam the curvature at position {{mvar|z}} is given by:

<math display="block">\frac{1}{R(z)} = \frac{z} {z^2 + z_\mathrm{R}^2} ,</math>

so the radius of curvature {{math|''R''(''z'')}} is <ref name="svelto153" />
<math display="block">R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right].</math>
Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.

===Elliptical and astigmatic beams===
Many laser beams have an elliptical cross-section.  Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams.  These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for {{mvar|x}} and {{mvar|y}} and distinct definitions of the {{math|1=''z'' = 0}} point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range {{math|±''π''/4}} contributed by each dimension.

An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.

===Gaussian as a decomposition into modes===

Arbitrary solutions of the [[Helmholtz equation#Paraxial approximation|paraxial Helmholtz equation]] can be decomposed as the sum of [[#Hermite-Gaussian modes|Hermite–Gaussian modes]] (whose amplitude profiles are separable in {{mvar|x}} and {{mvar|y}} using [[Cartesian coordinates]]), [[#Laguerre-Gaussian modes|Laguerre–Gaussian modes]] (whose amplitude profiles are separable in {{mvar|r}} and {{mvar|θ}} using [[cylindrical coordinates]]) or similarly as combinations of  [[#Ince-Gaussian modes|Ince–Gaussian modes]] (whose amplitude profiles are separable in {{mvar|ξ}} and {{mvar|η}} using [[elliptical coordinates]]).<ref name="siegman642">Siegman, p. 642.</ref><ref name="goubau">probably first considered by Goubau and Schwering (1961).</ref><ref name="ince-beams">Bandres and Gutierrez-Vega (2004)</ref> At any point along the beam {{mvar|z}} these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different [[#Gouy phase|Gouy phase]] which is why the net transverse profile due to a [[Superposition principle|superposition]] of modes evolves in {{mvar|z}}, whereas the propagation of any ''single'' Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.

Although there are other [[Transverse mode|modal decompositions]], Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is ''not'' operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's [[laser resonator|resonator]] (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM<sub>00</sub>) Gaussian mode.