Editing Gaussian beam

{{short description|Monochrome light beam whose amplitude envelope is a Gaussian function}}

[[Image:Gaussian beam w40mm lambda30mm.png|thumb|right|''Instantaneous'' absolute value of the real part of electric field amplitude of a TEM<sub>00</sub> gaussian beam, focal region. Showing <math>|\mathcal{Re}(E(t_1))|</math> thus with ''two'' peaks for each positive [[wavefront]].]]
[[Image:Laser gaussian profile.svg|thumb|right|Top: transverse intensity profile of a Gaussian beam that is propagating out of the page. Blue curve: electric (or magnetic) field amplitude vs. radial position from the beam axis. The black curve is the corresponding intensity.]]
[[Image:Green laser pointer TEM00 profile.JPG|thumb|right|A 5 mW green laser pointer beam, showing the TEM<sub>00</sub> profile]]

In [[optics]], a '''Gaussian beam''' is an idealized [[Light beam|beam]] of [[electromagnetic radiation]] whose [[Envelope (waves)|amplitude envelope]] in the transverse plane is given by a [[Gaussian function]]; this also implies a Gaussian [[irradiance|intensity]] (irradiance) profile. This fundamental (or TEM<sub>00</sub>) [[transverse mode|transverse]] Gaussian mode describes the intended output of many [[laser]]s, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal [[lens (optics)|lens]], a new Gaussian beam is produced. The [[Electric field|electric]] and [[magnetic field]] amplitude profiles along a circular Gaussian beam of a given [[wavelength]] and [[polarization (waves)|polarization]] are determined by two parameters: the [[#Beam waist|waist]] {{math|''w''<sub>0</sub>}}, which is a measure of the width of the beam at its narrowest point, and the position {{mvar|z}} relative to the waist.<ref name="svelto153">Svelto, pp. 153–5.</ref>

Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size ''w''(''z'') of the beam.

Fundamentally, the Gaussian is a solution of the paraxial [[Helmholtz equation]], the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.

==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.

== 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 | url-access=subscription }}</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"/>

==Power and intensity==

=== Power through an aperture ===
With a beam centered on an [[aperture]], the [[power (physics)|power]] {{mvar|P}} passing through a circle of radius {{mvar|r}} in the transverse plane at position {{mvar|z}} is<ref name="melles griot">{{Cite web |url=http://www.pa.msu.edu/courses/2010fall/phy431/PostNotes/PHY431-Notes-GaussianBeamOptics.pdf |title=Melles Griot. Gaussian Beam Optics |access-date=2015-04-07 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304031525/http://www.pa.msu.edu/courses/2010fall/phy431/PostNotes/PHY431-Notes-GaussianBeamOptics.pdf |url-status=dead }}</ref>
<math display="block">P(r,z) =  P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right],</math>
where
<math display="block">P_0 = \frac{ 1 }{ 2 } \pi I_0 w_0^2</math>
is the total power transmitted by the beam.

For a circle of radius {{math|1=''r'' = ''w''(''z'')}}, the fraction of power transmitted through the circle is
<math display="block">\frac{P(z)}{P_0} = 1 - e^{-2} \approx 0.865.</math>

Similarly, about 90% of the beam's power will flow through a circle of radius {{math|1=''r'' = 1.07 × ''w''(''z'')}}, 95% through a circle of radius {{math|1=''r'' = 1.224 × ''w''(''z'')}}, and 99% through a circle of radius {{math|1=''r'' = 1.52 × ''w''(''z'')}}.<ref name="melles griot"/>

=== Peak intensity ===
The peak intensity at an axial distance {{mvar|z}} from the beam waist can be calculated as the limit of the enclosed power within a circle of radius {{mvar|r}}, divided by the area of the circle {{math|''πr''<sup>2</sup>}} as the circle shrinks:
<math display="block">I(0,z) = \lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} .</math>

The limit can be evaluated using [[L'Hôpital's rule]]:
<math display="block">I(0,z)
         = \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)} 
         = {2P_0 \over \pi w^2(z)} .</math>

==Complex beam parameter==
{{main article|Complex beam parameter}}
The spot size and curvature of a Gaussian beam as a function of {{mvar|z}} along the beam can also be encoded in the complex beam parameter {{math|''q''(''z'')}}<ref name="siegman638">Siegman, pp. 638–40.</ref><ref name="garg168">Garg, pp. 165–168.</ref> given by:
<math display="block"> q(z) =  z + iz_\mathrm{R} .</math>

The reciprocal of {{math|''q''(''z'')}} contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:<ref name="siegman638" />

<math display="block">{1 \over q(z)} = {1 \over R(z)} - i {\lambda \over n \pi w^2(z)} .</math>

The complex beam parameter simplifies the mathematical analysis of Gaussian beam propagation, and especially in the analysis of [[optical cavity|optical resonator cavities]] using [[ray transfer matrix analysis|ray transfer matrices]].

Then using this form, the earlier equation for the electric (or magnetic) field is greatly simplified. If we call {{mvar|u}} the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the {{mvar|x}} and {{mvar|y}} directions) then it can be separated in {{mvar|x}} and {{mvar|y}} according to:
<math display="block">u(x,y,z) = u_x(x,z)\, u_y(y,z) ,</math>

where 
<math display="block">\begin{align}
u_x(x,z) &= \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right), \\
u_y(y,z) &= \frac{1}{\sqrt{{q}_y(z)}} \exp\left(-i k \frac{y^2}{2 {q}_y(z)}\right),
\end{align}</math>

where {{math|''q''<sub>''x''</sub>(''z'')}} and {{math|''q''<sub>''y''</sub>(''z'')}} are the complex beam parameters in the {{mvar|x}} and {{mvar|y}} directions.

For the common case of a [[circular symmetry|circular beam profile]], {{math|''q''<sub>''x''</sub>(''z'') {{=}} ''q''<sub>''y''</sub>(''z'') {{=}} ''q''(''z'')}} and {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = ''r''<sup>2</sup>}}, which yields<ref>See Siegman (1986) p. 639. Eq. 29</ref>
<math display="block">u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) .</math>

==Beam optics==
[[File:Gaussian Beam and Lens Diagram.svg|thumb|A diagram of a gaussian beam passing through a lens.]]

When a gaussian beam propagates through a [[thin lens]], the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens, and that the lens is larger than the width of the beam.  The focal length of the lens <math>f</math>, the beam waist radius <math>w_0</math>, and beam waist position <math>z_0</math> of the incoming beam can be used to determine the beam waist radius <math>w_0'</math> and  position <math>z_0'</math> of the outgoing beam.

===Lens equation===
As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the [[Phase (waves)|phase]] that is added to each point <math>(x,y)</math> of the gaussian beam as it travels through the lens.<ref name="fourier derivation of gaussian lens">{{cite book | title = Fundamentals of Photonics | last1 = Saleh |first1=Bahaa E. A. |last2=Teich |first2=Malvin Carl  | publisher = John Wiley & Sons | location = New York |  year = 1991 | isbn= 0-471-83965-5 }} Chapter 3, "Beam Optics"</ref>  An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam [[wavefront]]s.<ref name="wavefront derivation of gaussian lens">{{cite journal |last1=Self |first1=Sidney |date= 1 March 1983|title= Focusing of spherical Gaussian beams|url=https://doi.org/10.1364/AO.22.000658 |journal=Applied Optics |volume=22 |issue=5 |pages=658–661 |doi=10.1364/AO.22.000658|pmid=18195851 |bibcode=1983ApOpt..22..658S |url-access=subscription }}</ref>

The exact solution to the above problem is expressed simply in terms of the magnification <math>M</math>

:<math>
\begin{align}
w_0' &= Mw_0\\[1.2ex]
(z_0'-f) &= M^2(z_0-f).
\end{align}
</math>

The magnification, which depends on <math>w_0</math> and <math>z_0</math>, is given by

:<math>
M = \frac{M_r}{\sqrt{1+r^2}}
</math>

where

:<math>
r = \frac{z_R}{z_0-f}, \quad M_r = \left|\frac{f}{z_0-f}\right|.
</math>

An equivalent expression for the beam position <math>z_0'</math> is

:<math>
\frac{1}{z_0+\frac{z_R^2}{(z_0-f)}}+\frac{1}{z_0'} = \frac{1}{f}.
</math>

This last expression makes clear that the ray optics [[Thin lens|thin lens equation]] is recovered in the limit that <math>\left|\left(\tfrac{z_R}{z_0}\right)\left(\tfrac{z_R}{z_0-f}\right)\right|\ll 1</math>.  It can also be noted that if <math>\left|z_0+\frac{z_R^2}{z_0-f}\right|\gg f</math> then the incoming beam is "well collimated" so that <math>z_0'\approx f</math>.

===Beam focusing===
In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot.  Mathematically, this implies minimization of the magnification <math>M</math>.  If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing <math>z_R</math> and minimizing <math>f</math>.  In this situation, it is justifiable to make the approximation <math>z_R^2/(z_0-f)^2\gg 1</math>, implying that <math>M\approx f/z_R</math> and yielding the result <math>w_0'\approx fw_0/z_R</math>.  This result is often presented in the form

:<math>
\begin{align}
2w_0' &\approx \frac{4}{\pi}\lambda F_\# \\[1.2ex]
z_0' &\approx f
\end{align}
</math>

where
:<math>
F_\# = \frac{f}{2w_0},
</math>

which is found after assuming that the medium has index of refraction <math>n\approx 1</math> and substituting <math>z_R=\pi w_0^2/\lambda</math>.  The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters <math>2w_0'</math> and <math>2w_0</math>, rather than the waist radii <math>w_0'</math> and <math>w_0</math>.

==Wave equation==
As a special case of [[electromagnetic radiation]], Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the [[electromagnetic wave equation|wave equation for an electromagnetic field]] in free space or in a homogeneous dielectric medium,<ref name="svelto148">Svelto, pp. 148–9.</ref> obtained by combining Maxwell's equations for the curl of {{mvar|E}} and the curl of {{mvar|H}}, resulting in: 
<math display="block"> \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},</math>
where {{mvar|c}} is the speed of light ''in the medium'', and {{mvar|U}} could either refer to the electric or magnetic field vector, as any specific solution for either determines the other. The Gaussian beam solution is valid only in the [[paraxial]] approximation, that is, where wave propagation is limited to directions within a small angle of an axis. Without loss of generality let us take that direction to be the {{math|+''z''}} direction in which case the solution {{mvar|U}} can generally be written in terms of {{mvar|u}} which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the [[wavenumber]] {{mvar|k}} in the {{mvar|z}} direction:<ref name="svelto148" />
<math display="block"> U(x, y, z, t) = u(x, y, z) e^{-i(kz-\omega t)} \, \hat{\mathbf x} \, .</math>

Using this form along with the paraxial approximation, {{math|∂<sup>2</sup>''u''/∂''z''<sup>2</sup>}} can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation ({{mvar|z}}), we have without loss of generality considered the polarization to be in the {{mvar|x}} direction so that we now solve a scalar equation for {{math|''u''(''x'', ''y'', ''z'')}}.

Substituting this solution into the wave equation above yields the [[Helmholtz equation#Paraxial approximation|paraxial approximation]] to the scalar wave equation:<ref name="svelto148" />
<math display="block">\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.</math>
Writing the wave equations in the [[light-cone coordinates]] returns this equation without utilizing any approximation.<ref>{{Cite journal |last1=Esarey |first1=E. |last2=Sprangle |first2=P. |last3=Pilloff |first3=M. |last4=Krall |first4=J. |date=1995-09-01 |title=Theory and group velocity of ultrashort, tightly focused laser pulses |url=https://opg.optica.org/josab/abstract.cfm?uri=josab-12-9-1695 |journal=JOSA B |language=EN |volume=12 |issue=9 |pages=1695–1703 |doi=10.1364/JOSAB.12.001695 |bibcode=1995JOSAB..12.1695E |issn=1520-8540|url-access=subscription }}</ref> Gaussian beams of any beam waist {{math|''w''<sub>0</sub>}} satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at {{mvar|z}} in terms of the complex beam parameter {{math|''q''(''z'')}} as defined above. There are many other solutions. As solutions to a [[linear system]], any combination of solutions (using addition or multiplication by a constant) is also a solution. The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above. In seeking paraxial solutions, and in particular ones that would describe laser radiation that is ''not'' in the fundamental Gaussian mode, we will look for families of solutions with gradually increasing products of their divergences and minimum spot sizes. Two important orthogonal decompositions of this sort are the Hermite–Gaussian or Laguerre-Gaussian modes, corresponding to rectangular and circular symmetry respectively, as detailed in the next section. With both of these, the fundamental Gaussian beam we have been considering is the lowest order mode.

==Higher-order modes==
{{see also|Transverse mode}}

=== Hermite-Gaussian modes ===
<!--Hermite-Gaussian mode redirects here-->
[[Image:Hermite-gaussian.png|thumb|right|Twelve Hermite-Gaussian modes]]
It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called ''Hermite-Gaussian modes'', any of which are given by the product of a factor in {{mvar|x}} and a factor in {{mvar|y}}. Such a solution is possible due to the separability in {{mvar|x}} and {{mvar|y}} in the [[paraxial Helmholtz equation]] as written in [[Cartesian coordinate system|Cartesian coordinates]].<ref>Siegman (1986), p645, eq. 54</ref> Thus given a mode of order {{math|(''l'', ''m'')}} referring to the {{mvar|x}} and {{mvar|y}} directions, the electric field amplitude at {{math|''x'', ''y'', ''z''}} may be given by:
<math display="block"> E(x,y,z) = u_l(x,z) \, u_m(y,z) \, \exp(-ikz), </math> 
where the factors for the {{mvar|x}} and {{mvar|y}} dependence are each given by:
<math display="block">
u_J(x,z) = \left(\frac{\sqrt{2/\pi}}{ 2^J \, J! \; w_0}\right)^{\!\!1/2} \!\! \left( \frac{{q}_0}{{q}(z)}\right)^{\!\!1/2} \!\! \left(- \frac{{q}^\ast(z)}{{q}(z)}\right)^{\!\! J/2} \!\! H_J\!\left(\frac{\sqrt{2}x}{w(z)}\right)  \, \exp \left(\! -i \frac{k x^2}{2 {q}(z)}\right) ,
</math>
where we have employed the complex beam parameter {{math|''q''(''z'')}} (as defined above) for a beam of waist {{math|''w''<sub>0</sub>}} at {{mvar|z}} from the focus. In this form, the first factor is just a normalizing constant to make the set of {{math|''u<sub>J</sub>''}} [[orthonormal]]. The second factor is an additional normalization dependent on {{mvar|z}} which compensates for the expansion of the spatial extent of the mode according to {{math|''w''(''z'')/''w''<sub>0</sub>}} (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders {{mvar|J}}.

The final two factors account for the spatial variation over {{mvar|x}} (or {{mvar|y}}). The fourth factor is the [[Hermite polynomial]] of order {{mvar|J}} ("physicists' form", i.e. {{math|1=''H''<sub>1</sub>(''x'') = 2''x''}}), while the fifth accounts for the Gaussian amplitude fall-off {{math|exp(−''x''<sup>2</sup>/''w''(''z'')<sup>2</sup>)}}, although this isn't obvious using the complex {{mvar|q}} in the exponent. Expansion of that exponential also produces a phase factor in {{mvar|x}} which accounts for the wavefront curvature ({{math|1/''R''(''z'')}}) at {{mvar|z}} along the beam.

Hermite-Gaussian modes are typically designated "TEM<sub>''lm''</sub>"; the fundamental Gaussian beam may thus be referred to as TEM<sub>00</sub> (where ''TEM'' is ''[[transverse mode|transverse electro-magnetic]]''). Multiplying {{math|''u<sub>l</sub>''(''x'', ''z'')}} and {{math|''u<sub>m</sub>''(''y'', ''z'')}} to get the 2-D mode profile, and removing the normalization so that the leading factor is just called {{math|''E''<sub>0</sub>}}, we can write the {{math|(''l'', ''m'')}} mode in the more accessible form:

<math display="block">\begin{align}
  E_{l, m}(x, y, z) ={}
    & E_0 \frac{w_0}{w(z)}\, H_l \!\Bigg(\frac{\sqrt{2} \,x}{w(z)}\Bigg)\, H_m \!\Bigg(\frac{\sqrt{2} \,y}{w(z)}\Bigg) \times {}  \exp \left( {-\frac{x^2+y^2}{w^2(z)}} \right) \exp \left( {-i\frac{k(x^2 + y^2)}{2R(z)}} \right) \times {}  \exp \big(i \psi(z)\big) \exp(-ikz).
\end{align}</math>

In this form, the parameter {{math|''w''<sub>0</sub>}}, as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at {{math|1=''z'' = 0}}. Given that {{math|''w''<sub>0</sub>}}, {{math|''w''(''z'')}} and {{math|''R''(''z'')}} have the same definitions as for the fundamental Gaussian beam described [[#Evolving beam width|above]]. It can be seen that with {{math|1=''l'' = ''m'' = 0}} we obtain the fundamental Gaussian beam described earlier (since {{math|1=''H''<sub>0</sub> = 1}}). The only specific difference in the {{mvar|x}} and {{mvar|y}} profiles at any {{mvar|z}} are due to the Hermite polynomial factors for the order numbers {{mvar|l}} and {{mvar|m}}. However, there is a change in the evolution of the modes' Gouy phase over {{mvar|z}}:
<math display="block"> \psi(z)  = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right), </math>

where the combined order of the mode {{mvar|N}} is defined as {{math|1=''N'' = ''l'' + ''m''}}. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by {{math|±''π''/2}} radians over all of {{mvar|z}} (and only by {{math|±''π''/4}} radians between {{math|±''z''<sub>R</sub>}}), this is increased by the factor {{math|''N'' + 1}} for the higher order modes.<ref name="gouy_phase_shift">{{cite encyclopedia  |title=Gouy Phase Shift |url=https://www.rp-photonics.com/gouy_phase_shift.html |encyclopedia=Encyclopedia of Laser Physics and Technology |publisher=RP Photonics |first=Rüdiger |last=Paschotta |date=12 December 2006 |access-date=May 2, 2014}}</ref>

Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.

=== Laguerre-Gaussian modes ===
<!--Several terms redirect here.-->
[[Image:Intensity profiles of Laguerre-Gaussian modes.pdf|thumb|right|Intensity profiles of the first 12 Laguerre-Gaussian modes.]]
Beam profiles which are circularly symmetric (or lasers with cavities that are  cylindrically symmetric) are often best solved using the  Laguerre-Gaussian modal decomposition.<ref name="goubau"/> These functions are written in [[cylindrical coordinates]] using [[Laguerre polynomials#Generalized Laguerre polynomials|generalized Laguerre polynomials]]. Each transverse mode is again labelled using two integers, in this case the radial index {{math|''p'' ≥ 0}} and the azimuthal index {{mvar|l}} which can be positive or negative (or zero):<ref name="orbital momentum of light">{{cite journal |title=On the properties of circular beams: normalization, Laguerre–Gauss expansion, and free-space divergence |journal=Optics Letters |volume=40 |issue=8 |date=April 8, 2015 |pages=1717–1720 |url=https://doi.org/10.1364/OL.40.001717 |first1=G. |last1=Vallone |doi=10.1364/OL.40.001717|pmid=25872056 |arxiv=1501.07062 |bibcode=2015OptL...40.1717V |s2cid=36312938 }}</ref><ref>{{Cite journal |last1=Miatto |first1=Filippo M. |last2=Yao |first2=Alison M. |last3=Barnett |first3=Stephen M. |date=2011-03-15 |title=Full characterization of the quantum spiral bandwidth of entangled biphotons |url=https://link.aps.org/doi/10.1103/PhysRevA.83.033816 |journal=Physical Review A |language=en |volume=83 |issue=3 |page=033816 |doi=10.1103/PhysRevA.83.033816 |issn=1050-2947|arxiv=1011.5970 |bibcode=2011PhRvA..83c3816M }}</ref>
[[File:Focused Laguerre-Gaussian beam.webm|thumb|right|A Laguerre-Gaussian beam with l=1 and p=0]] 
<math display="block">\begin{align}
  u(r, \phi, z) ={}
    &C^{LG}_{lp}\frac{1}{w(z)}\left(\frac{r \sqrt{2}}{w(z)}\right)^{\! |l|} \exp\! \left(\! -\frac{r^2}{w^2(z)}\right)L_p^{|l|}  \! \left(\frac{2r^2}{w^2(z)}\right) \times {} \\
    &\exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \exp(-i l \phi) \, \exp(i \psi(z)) ,
\end{align}</math>

where {{math|''L<sub>p</sub><sup>l</sup>''}} are the [[Laguerre polynomials#Generalized Laguerre polynomials|generalized Laguerre polynomials]]. {{math|''C{{su|p=LG|b=lp}}''}} is a required normalization constant:<ref name="LG_normalization">Note that the normalization used here (total intensity for a fixed {{math|''z''}} equal to unity) differs from that used in section [[#Mathematical form]] for the Gaussian mode. For {{math|1=''l = p = 0''}} the Laguerre-Gaussian mode reduces to the standard Gaussian mode, but due to different normalization conditions the two formulas do not coincide.</ref>
<math display="block">C^{LG}_{lp} = \sqrt{\frac{2 p!}{\pi(p+|l|)!}} \Rightarrow \int_0^{2\pi}d\phi\int_0^\infty dr\; r \,|u(r,\phi,z)|^2=1,</math>.

{{math|''w''(''z'')}} and {{math|''R''(''z'')}} have the same definitions as [[#Beam parameters|above]]. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor {{math|''N'' + 1}}:
<math display="block">\psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right) ,</math>
where in this case the combined mode number {{math|1=''N'' = {{mabs|''l''}} + 2''p''}}. As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in {{mvar|r}} but now multiplied by a Laguerre polynomial. The effect of the [[rotational modes|rotational mode]] number {{mvar|l}}, in addition to affecting the Laguerre polynomial, is mainly contained in the ''phase'' factor {{math|exp(−''ilφ'')}}, in which the beam profile is advanced (or retarded) by {{mvar|l}} complete {{math|2''π''}} phases in one rotation around the beam (in {{mvar|φ}}). This is an example of an [[optical vortex]] of topological charge {{mvar|l}}, and can be associated with the [[orbital angular momentum of light]] in that mode.

=== Ince-Gaussian modes ===
[[File:Ince Gaussian Modes.jpg|thumb|Transverse amplitude profile of the lowest order even Ince-Gaussian modes.]]
In [[elliptic coordinates]], one can write the higher-order modes using [[Ince polynomial]]s. The even and odd Ince-Gaussian modes are given by<ref name="ince-beams"/>

<math display="block">
u_\varepsilon \left( \xi ,\eta ,z\right) = \frac{w_{0}}{w\left(
z\right) }\mathrm{C}_{p}^{m}\left( i\xi ,\varepsilon \right) \mathrm{C}
_{p}^{m}\left( \eta ,\varepsilon \right) \exp \left[ -ik\frac{r^{2}}{
2q\left( z\right) }-\left( p+1\right) \zeta\left( z\right) \right] ,
</math>
where {{mvar|ξ}} and {{mvar|η}} are the radial and angular elliptic coordinates defined by
<math display="block">\begin{align}
x &= \sqrt{\varepsilon /2}\;w(z) \cosh \xi \cos \eta ,\\
y &= \sqrt{\varepsilon /2}\;w(z) \sinh \xi \sin \eta .
\end{align}</math>
{{math|''C{{su|b=p|p=m}}''(''η'', ''ε'')}} are the even Ince polynomials of order {{mvar|p}} and degree {{mvar|m}} where {{mvar|ε}} is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for {{math|1=''ε'' = ∞}} and {{math|1=''ε'' = 0}} respectively.<ref name=ince-beams/>

=== Hypergeometric-Gaussian modes ===

There is another important class of paraxial wave modes in [[cylindrical coordinates]] in which the [[complex amplitude]] is proportional to a [[confluent hypergeometric function]].

These modes have a [[Mathematical singularity|singular]] phase profile and are [[eigenfunction]]s of the [[photon orbital angular momentum]]. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate {{math|1=''ρ'' = ''r''/''w''<sub>0</sub>}} and the normalized longitudinal coordinate {{math|1=''Ζ'' = ''z''/''z''<sub>R</sub>}} as follows:<ref name="Karimi et al. 2007">Karimi et al. (2007)</ref>

<math display="block">\begin{align}
  u_{\mathsf{p}m}(\rho, \phi, \Zeta) {}={}
    &\sqrt{\frac{2^{\mathsf{p} + |m| + 1}}{\pi\Gamma(\mathsf{p} + |m| + 1)}}\; \frac{\Gamma\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}{\Gamma(|m| + 1)}\, i^{|m|+1} \times{} \\
    &\Zeta^{\frac{\mathsf{p}}{2}}\, (\Zeta + i)^{-\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}\, \rho^{|m|} \times{} \\
    &\exp\left(-\frac{i\rho^2}{\Zeta + i}\right)\, e^{im\phi}\, {}_1F_1 \left(-\frac{\mathsf{p}}{2}, |m| + 1; \frac{\rho^2}{\Zeta(\Zeta + i)}\right)
\end{align}</math>

where the rotational index {{mvar|m}} is an integer, and <math> {\mathsf p}\ge-|m| </math> is real-valued, {{math|Γ(''x'')}} is the gamma function and {{math|<sub>1</sub>''F''<sub>1</sub>(''a'', ''b''; ''x'')}} is a confluent hypergeometric function.

Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes,<ref name="Karimi et al. 2007"/> and the modified Laguerre–Gaussian modes.

The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist ({{math|1=''z'' = 0}}):
<math display="block">u(\rho, \phi, 0) \propto \rho^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}.</math>

==See also==
* [[Bessel beam]]
* [[Tophat beam]]
* [[Laser beam profiler]]
* [[Quasioptics]]

==Notes==
<references/>

==References==
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*{{cite arXiv
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*{{cite book | title = Fundamentals of Photonics | last1 = Saleh |first1=Bahaa E. A. |last2=Teich |first2=Malvin Carl  | publisher = John Wiley & Sons | location = New York |  year = 1991 | isbn= 0-471-83965-5 }} Chapter 3, "Beam Optics," pp.&nbsp;80–107.
*{{cite book | first = Anthony E.|last=Siegman|year=1986|title=Lasers| url = https://archive.org/details/lasers0000sieg| url-access = registration|publisher=University Science Books|isbn= 0-935702-11-3}} Chapter 16.
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==External links==
* [https://www.newport.com/n/gaussian-beam-optics Gaussian Beam Optics Tutorial, Newport]

{{Lasers}}

[[Category:Physical optics]]
[[Category:Laser science]]
[[Category:Electromagnetic radiation]]