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| [[Image:Gaussian beam w40mm lambda30mm.png|thumb|right|Instantaneous intensity of a Gaussian beam.]]
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| [[Image:Green laser pointer TEM00 profile.JPG|thumb|right|A 5 mW green laser pointer beam profile, showing the TEM<sub>00</sub> profile]]
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| In [[optics]], a '''Gaussian beam''' is a [[Light beam|beam]] of [[electromagnetic radiation]] whose transverse [[electric field]] and [[irradiance|intensity]] (irradiance) distributions are well approximated by [[Gaussian function]]s. Many [[laser]]s emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the ''fundamental [[transverse mode]]'', or "TEM<sub>00</sub> mode" of the laser's [[optical resonator]]. When [[refraction|refracted]] by a [[Diffraction-limited system|diffraction-limited]] [[lens (optics)|lens]], a Gaussian beam is transformed into another Gaussian beam (characterized by a different set of parameters), which explains why it is a convenient, widespread [[mathematical models in physics|model]] in laser optics.
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| The mathematical function that describes the Gaussian beam is a solution to the [[paraxial approximation|paraxial]] form of the [[Helmholtz equation#Paraxial approximation|Helmholtz equation]]. The solution, in the form of a Gaussian function, represents the [[complex number|complex]] amplitude of the beam's [[electric field]]. The electric field and [[magnetic field]] together propagate as an [[electromagnetic wave]]. A description of just one of the two fields is sufficient to describe the properties of the beam.
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| | | 相关的主题文章： |
| The behavior of the field of a Gaussian beam as it propagates is described by a few parameters such as the spot size, the radius of curvature, and the Gouy phase.<ref name="svelto153">Svelto, pp. 153–5.</ref>
| | <ul> |
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| Other solutions to the paraxial form of the Helmholtz equation exist. Solving the equation in Cartesian coordinates leads to a family of solutions known as the Hermite–Gaussian modes, while solving the equation in cylindrical coordinates leads to the Laguerre–Gaussian modes.<ref name="siegman642">Siegman, p. 642.</ref><ref name="goubau">probably first considered by Goubau and Schwering (1961).</ref> For both families, the lowest-order solution describes a Gaussian beam, while higher-order solutions describe higher-order transverse modes in an optical resonator.
| | <li>[http://www.tc139.cn/news/html/?214241.html http://www.tc139.cn/news/html/?214241.html]</li> |
| [[Image:Laser gaussian profile.svg|thumb|right|The top portion of the diagram shows the two-dimensional intensity profile of a Gaussian beam that is propagating out of the page. The blue curve, below, is a plot of the electric field amplitude as a function of distance from the center of the beam. The black curve is the corresponding intensity function.]]
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| | | <li>[http://www.ticketlodge.com/ticket-lodge-blog/14-06-17/Jesse_McCartney_In_Technicolor_Tour.aspx/ http://www.ticketlodge.com/ticket-lodge-blog/14-06-17/Jesse_McCartney_In_Technicolor_Tour.aspx/]</li> |
| ==Mathematical form==
| | |
| The Gaussian beam is a [[transverse electromagnetic mode|transverse electromagnetic (TEM) mode]].<ref name="svelto158">Svelto, p. 158.</ref> A mathematical expression for its complex electric field amplitude can be found by solving the paraxial Helmholtz equation, yielding<ref name="svelto153" />
| | <li>[http://www.film-video-dvd-production.com/spip.php?article6/ http://www.film-video-dvd-production.com/spip.php?article6/]</li> |
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| :<math>E(r,z) = E_0 \frac{w_0}{w(z)} \exp \left( \frac{-r^2}{w(z)^2} -ikz -ik \frac{r^2}{2R(z)} +i \zeta(z) \right)\ , </math>
| | <li>[http://www.winnerwayx.com/forum.php?mod=viewthread&tid=2020 http://www.winnerwayx.com/forum.php?mod=viewthread&tid=2020]</li> |
| | | |
| where<ref name="svelto153" />
| | <li>[http://www.vozesdaeducacao.org.br/bbpress/profile.php?id=472089 http://www.vozesdaeducacao.org.br/bbpress/profile.php?id=472089]</li> |
| :<math>r</math> is the radial distance from the center axis of the beam,
| | |
| :<math>z</math> is the axial distance from the beam's narrowest point (the "waist"),
| | </ul> |
| :<math>i</math> is the [[imaginary unit]] (for which <math>i^2 = -1</math>),
| |
| :<math> k = 2 \pi/\lambda</math> is the [[wave number]] (in [[radian]]s per meter),
| |
| :<math>E_0 = |E(0,0)|</math>,
| |
| :<math>w(z)</math> is the radius at which the field amplitude and intensity drop to 1/''e'' and 1/''e''<sup>2</sup> of their axial values, respectively,
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| :<math>w_0 = w(0)</math> is the [[#Beam width or spot size|waist size]],
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| :<math>R(z)</math> is the [[#Radius of curvature|radius of curvature]] of the beam's wavefronts, and
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| :<math>\zeta(z)</math> is the [[#Gouy phase|Gouy phase shift]], an extra contribution to the phase that is seen in Gaussian beams.
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| Additionally, the field has a time dependence factor <math>e^{i\omega t}</math> that has been suppressed in the above expression.
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| The corresponding time-averaged [[intensity (physics)|intensity]] (or [[irradiance]]) distribution is
| |
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| :<math>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^2(z)} \right)\ , </math> | |
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| where <math>I_0 = I(0,0)</math> is the intensity at the center of the beam at its waist. The constant <math>\eta \,</math> is the [[Wave impedance|characteristic impedance]] of the medium in which the beam is propagating. For free space, <math> \eta = \eta_0 = \sqrt{\mu_0/\varepsilon_0} = 1/(\varepsilon_0 c) \approx 376.7 \ \Omega</math>.
| |
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| == Beam parameters == | |
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| The geometry and behavior of a Gaussian beam are governed by a set of '''beam parameters''', which are defined in the following sections.
| |
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| ===Beam width or spot size===<!--Beam waist redirects here-->
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| {{see also|Beam diameter}}
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| [[Image:GaussianBeamWaist.svg|thumb|350px|right|Gaussian beam width ''w''(''z'') as a function of the axial distance ''z''. ''w''<sub>0</sub>: beam waist; ''b'': depth of focus; ''z''<sub>R</sub>: [[Rayleigh range]]; <math>\Theta</math>: total angular spread]]
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| For a Gaussian beam propagating in free space, the spot size (radius) ''w''(''z'') will be at a minimum value ''w''<sub>0</sub> at one place along the beam axis, known as the ''beam waist''. For a beam of [[wavelength]] λ at a distance ''z'' along the beam from the beam waist, the variation of the spot size is given by<ref name="svelto153" />
| |
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| :<math>w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } \ . </math>
| |
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| where the origin of the z-axis is defined, without loss of generality, to coincide with the beam waist, and where<ref name="svelto153" />
| |
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| :<math>z_\mathrm{R} = \frac{\pi w_0^2}{\lambda}</math>
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| is called the [[Rayleigh range]].
| |
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| ===Rayleigh range and confocal parameter===
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| At a distance from the waist equal to the Rayleigh range ''z''<sub>R</sub>, the width ''w'' of the beam is<ref name="svelto153" />
| |
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| :<math> w(\pm z_\mathrm{R}) = \sqrt{2} w_0.</math>
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| The distance between these two points is called the ''confocal parameter'' or ''depth of focus'' of the beam:
| |
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| :<math>b = 2 z_\mathrm{R} = \frac{2 \pi w_0^2}{\lambda}\,.</math>
| |
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| === Radius of curvature ===
| |
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| ''R''(''z'') is the ''[[Radius of curvature (optics)|radius of curvature]]'' of the wavefronts comprising the beam. Its value as a function of position is<ref name="svelto153" />
| |
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| :<math>R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right] \ . </math>
| |
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| ===Beam divergence===
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| The parameter <math>w(z)</math> increases linearly with <math>z</math> for <math>z \gg z_\mathrm{R}</math>. This means that far from the waist, the beam is cone-shaped. The angle between the straight line <math>r=w(z)</math> and the central axis of the beam (<math>r=0</math>) is called the ''divergence'' of the beam. It is given by<ref name="svelto153" />
| |
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| :<math>\theta \simeq \frac{\lambda}{\pi w_0} \qquad (\theta \mathrm{\ in\ radians}). </math>
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| The total angular spread of the beam far from the waist is then given by
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| :<math>\Theta = 2 \theta\ .</math>
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| Because the divergence is inversely proportional to the spot size, a Gaussian beam that is focused to a small spot spreads out rapidly as it propagates away from that spot. To keep a laser beam very well [[Collimated light|collimated]], it must have a large diameter. This relationship between beam width and divergence is due to [[diffraction]]. Non-Gaussian beams also exhibit this effect, but a Gaussian beam is a special case where the product of width and divergence is the smallest possible.
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| Since the gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the direction of propagation.<ref>Siegman (1986) p. 630.</ref> From the above expression for divergence, this means the Gaussian beam model is valid only for beams with waists larger than about <math>2\lambda/\pi</math>.
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| [[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_0</math>. 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 ''M''<sup>2</sup> ("[[M squared]]"). The ''M''<sup>2</sup> for a Gaussian beam is one. All real laser beams have ''M''<sup>2</sup> values greater than one, although very high quality beams can have values very close to one.
| |
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| ===Gouy phase===
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| The ''longitudinal phase delay'' or ''Gouy phase'' of the beam is<ref name="svelto153" />
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| :<math>\zeta(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right) \ .</math>
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| The Gouy phase indicates that as a Gaussian beam passes through a focus, it acquires an additional phase shift of π, in addition to the usual <math>e^{-ikz}</math> phase shift that would be expected from a plane wave.<ref name="svelto153" />
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| ===Complex beam parameter===
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| {{main|Complex beam parameter}}
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| Information about the spot size and radius of curvature of a Gaussian beam can be encoded in the complex beam parameter, <math>q(z)</math>:<ref name="siegman638">Siegman, pp. 638–40.</ref>
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| :<math> q(z) = z + q_0 = z + iz_\mathrm{R} \ .</math>
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| The reciprocal <math>1/q(z)</math> shows the relationship between <math>q(z)</math>, <math>w(z)</math>, and <math>R(z)</math> explicitly:<ref name="siegman638" />
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| :<math> { 1 \over q(z) } = { 1 \over z + iz_\mathrm{R} } = { z \over z^2 + z_\mathrm{R}^2 } - i { z_\mathrm{R} \over z^2 + z_\mathrm{R}^2 } = {1 \over R(z) } - i { \lambda \over \pi w^2(z) }.</math>
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| The complex beam parameter plays a key role in the analysis of Gaussian beam propagation, and especially in the analysis of [[optical cavity|optical resonator cavities]] using [[ray transfer matrix analysis|ray transfer matrices]].
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| In terms of the complex beam parameter <math>{q}</math>, a Gaussian field with one transverse dimension is proportional to
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| :<math> | |
| {u}(x,z) = \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right).
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| </math>
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| In two dimensions one can write the potentially elliptical or astigmatic beam as the product
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| :<math>
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| {u}(x,y,z) = {u}(x,z)\, {u}(y,z),
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| </math>
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| which for the common case of [[circular symmetry]] where <math>{q}_x = {q}_y = {q}</math> and <math>x^2 + y^2 = r^2</math> yields<ref>See Siegman (1986) p. 639. Eq. 29</ref>
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| :<math>
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| {u}(r,z) = \frac{1}{{q}(z)}\exp\left( -i k\frac{r^2}{2 {q}(z)}\right).
| |
| </math>
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| ==Power and intensity==
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| === Power through an aperture ===
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| The [[power (physics)|power]] ''P'' passing through a circle of radius ''r'' in the transverse plane at position ''z'' is
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| :<math> P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]\ ,</math>
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| where
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| :<math> P_0 = { 1 \over 2 } \pi I_0 w_0^2 </math>
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| is the total power transmitted by the beam.
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| For a circle of radius <math>r = w(z) \, </math>, the fraction of power transmitted through the circle is
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| :<math>{ P(z) \over P_0 } = 1 - e^{-2} \approx 0.865\ .</math>
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| Similarly, about 95 percent of the beam's power will flow through a circle of radius <math>r = 1.224\cdot w(z) \, </math>.
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| === Peak and average intensity ===
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| The peak intensity at an axial distance <math>z</math> from the beam waist is calculated using [[L'Hôpital's rule]] as the limit of the enclosed power within a circle of radius <math>r</math>, divided by the area of the circle <math>\pi r^2</math>:
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| :<math>I(0,z) =\lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2}
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| = \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)}
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| = {2P_0 \over \pi w^2(z)}. </math>
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| The peak intensity is thus exactly twice the ''average intensity'', obtained by dividing the total power by the area within the radius <math>w(z)</math>.
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| ==Derivation==
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| The Gaussian beam formalism begins with 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>
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| :<math> \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},</math>
| |
| where <math>U</math> may stand for any one of the six field components <math>E_x</math>, <math>E_y</math>, <math>E_z</math>, <math>B_x</math>, <math>B_y</math>, or <math>B_z</math>. The Gaussian beam formalism proceeds by writing down a solution of the form<ref name="svelto148" />
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| :<math> U(x,y,z,t) = u(x,y,z) e^{-i(kz-\omega t)},</math>
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| where it is assumed that the beam is sufficiently [[collimated]] along the <math>z</math> axis that <math>\partial^2 u/\partial z^2</math> may be neglected. Substituting this solution into the wave equation above yields the [[Helmholtz equation#Paraxial approximation|paraxial approximation]] to the wave equation:<ref name="svelto148" />
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| :<math>\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.</math>
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| Solving this differential equation yields an infinite set of functions, of which the Gaussian beam is the lowest-order solution or ''[[transverse mode|mode]]''.
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| ==Higher-order modes==
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| {{see also|Transverse mode}}
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| Gaussian beams are just one possible solution to the paraxial wave equation. Various other sets of [[orthogonal]] solutions are used for modelling laser beams. In the general case, if a complete [[Basis (linear algebra)|basis set]] of solutions is chosen, any real laser beam can be described as a superposition of solutions from this set. The design of the laser determines which basis set of solutions is most useful. In some cases the output of a laser may closely approximate a single higher-order mode. Hermite-Gaussian modes are particularly common, since many laser systems have Cartesian reflection symmetry in the plane perpendicular to the beam's propagation direction.
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| === Hermite-Gaussian modes === <!--Hermite-Gaussian mode redirects here-->
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| [[Image:Hermite-gaussian.png|thumb|250px|right|Twelve Hermite-Gaussian modes]]
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| Hermite-Gaussian modes are a convenient description for the output of lasers whose cavity design is not radially symmetric, but rather has a distinction between horizontal and vertical. In terms of the previously defined complex <math>q</math> parameter, the amplitude distribution in the <math>x</math>-plane is proportional to
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| :<math> | |
| {u}_n(x,z) = \left(\frac{2}{\pi}\right)^{1/4} \left(\frac{1}{2^n n! w_0}\right)^{1/2} \left( \frac{{q}_0}{{q}(z)}\right)^{1/2} \left[\frac{{q}_0}{{q}_0^\ast} \frac{{q}^\ast(z)}{{q}(z)}\right]^{n/2} H_n\left(\frac{\sqrt{2}x}{w(z)}\right) \exp\left[-i \frac{k x^2}{2 {q}(z)}\right]
| |
| </math>
| |
| where the function <math>H_n(x)</math> is the [[Hermite polynomial]] of order <math>n</math> (physicists' form, i.e. <math>H_1(x)=2x\,</math>), and the asterisk indicates [[complex conjugation]]. For the case <math>n=0</math> the equation yields a Gaussian transverse distribution.
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| For two-dimensional [[rectangular coordinates]] one constructs a function <math>{u}_{mn}(x,y,z)=u_m(x,z) u_n(y,z)</math>, where <math>u_n(y,z)</math> has the same form as <math>u_m(x,z)</math>. Mathematically this property is due to the [[separation of variables]] applied to the [[paraxial Helmholtz equation]] for [[Cartesian coordinate system|Cartesian coordinates]].<ref>Siegman (1986), p645, eq. 54</ref>
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| Hermite-Gaussian modes are typically designated "TEM<sub>''mn''</sub>", where ''m'' and ''n'' are the polynomial indices in the x and y directions. A Gaussian beam is thus TEM<sub>00</sub>.
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| === Laguerre-Gaussian modes === <!--Several terms redirect here.-->
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| [[Image:LG-wiki.jpg|thumb|right|upright=1.5|The intensity profiles of twelve Laguerre-Gaussian modes]]
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| If the problem is cylindrically symmetric, the natural solutions of the paraxial wave equation are Laguerre-Gaussian modes.<ref name="goubau">probably first considered by Goubau and Schwering (1961).</ref> They are written in [[cylindrical coordinates]] using [[Laguerre polynomial]]s
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| :<math>{u}(r,\phi,z)=\frac{C^{LG}_{lp}}{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) | |
| \exp\left( i k \frac{r^2}{2 R(z)}\right)\exp(i l \phi)\exp\left[i(2p+|l|+1)\zeta(z)\right],
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| </math>
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| where <math>L_p^l</math> are the generalized Laguerre polynomials, the radial index <math>p\ge 0</math> and the azimuthal index is <math>l</math>. <math>C^{LG}_{lp}</math> is an appropriate normalization constant; <math>w(z)</math>, <math>R(z)</math> and <math>\zeta(z)</math> are beam parameters defined [[#Beam parameters|above]].
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| === Ince-Gaussian modes ===
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| 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>Bandres and Gutierrez-Vega (2004)</ref>
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| :<math>
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| u_\varepsilon \left( \xi ,\eta ,z\right) = \frac{w_{0}}{w\left(
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| z\right) }\mathrm{C}_{p}^{m}\left( i\xi ,\varepsilon \right) \mathrm{C}
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| _{p}^{m}\left( \eta ,\varepsilon \right) \exp \left[ -ik\frac{r^{2}}{
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| 2q\left( z\right) }-\left( p+1\right) \psi _{GS}\left( z\right) \right] ,
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| </math>
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| where <math>\xi</math> and <math>\eta</math> are the radial and angular elliptic coordinates
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| defined by
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| :<math>
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| x = \sqrt{\varepsilon /2}w\left( z\right) \cosh \xi \cos \eta ,
| |
| </math>
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| :<math> | |
| y = \sqrt{\varepsilon /2}w\left( z\right) \sinh \xi \sin \eta.
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| </math>
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| <math>{C}_{p}^{m}\left( \eta ,\epsilon \right)</math> are the even Ince
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| polynomials of order <math>p</math> and degree <math>m</math>, <math>\varepsilon</math> is the ellipticity parameter, and <math>\psi _{GS}\left( z\right) =\arctan \left( z/z_\mathrm{R}\right)</math>
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| is the Gouy phase. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for <math>\varepsilon=\infty</math> and <math>\varepsilon=0</math> respectively.
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| | |
| === Hypergeometric-Gaussian modes ===
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| | |
| There is another important class of paraxial wave modes in [[polar coordinates]] in which the complex amplitude is proportional to a [[confluent hypergeometric function]].
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| These modes have a [[Mathematical singularity|singular]] phase profile and are eigenfunctions of the [[photon orbital angular momentum]]. The intensity profile is characterized by a single brilliant ring with a [[Mathematical singularity|singularity]] at its center, where the field amplitude vanishes.<ref>Karimi et. al (2007)</ref>
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| | |
| :<math> | |
| u_{pm}(\rho,\theta;\zeta)=
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| \sqrt{\frac{2^{p+|m|+1}}{\pi\Gamma(p+|m|+1)}} \frac{\Gamma(1+|m|+\frac{p}{2})}{\Gamma(|m|+1)}
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| \,\,i^{|m|+1}\zeta^{\frac{p}{2}}(\zeta+i)^{-(1+|m|+\frac{p}{2})}\rho^{|m|}e^{-\frac{i\rho^2}{(\zeta+i)}}e^{im\phi}{}_{1}F_{1}\left(-\frac{p}{2}, |m|+1;\frac{r^2}{\zeta(\zeta+i)}\right),
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| </math>
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| | |
| where <math> m </math> is integer, <math> p\ge-|m| </math> is real valued, <math> \Gamma(x) </math> is the gamma function and <math> {}_{1}F_{1}(a,b;x) </math> is a confluent hypergeometric
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| function.
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| Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre–Gaussian modes.
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| 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 pupil plane:
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| | |
| :<math>
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| u(\rho,\phi,0) \propto \rho^{p+|m|}e^{-\rho^2+im\phi}.
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| </math>
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| | |
| See [[Optical vortex]], which explains that the outcoming wave from a pitch-fork hologram is a sub-family of HyGG modes. The HyGG profile while beam propagates along <math>\zeta</math> has a dramatic change and it is not a stable mode below the [[Rayleigh range]].
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| | |
| ==See also==
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| * [[Bessel beam]]
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| * [[Tophat beam]]
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| * [[Laser beam profiler]]
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| | |
| ==Notes==
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| <references/> | |
| | |
| ==References==
| |
| *{{cite book | title = Fundamentals of Photonics | author = Saleh, Bahaa E. A. and Teich, Malvin Carl | publisher = John Wiley & Sons | location = New York | year = 1991 | isbn= 0-471-83965-5 }} Chapter 3, "Beam Optics," pp. 80–107.
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| | |
| *{{cite book | title = Optical Coherence and Quantum Optics | author = Mandel, Leonard and Wolf, Emil | publisher = Cambridge University Press | location = Cambridge | year = 1995 | isbn= 0-521-41711-2 }} Chapter 5, "Optical Beams," pp. 267.
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| | |
| *{{cite book | first = Anthony E.|last=Siegman|year=1986|title=Lasers|publisher=University Science Books|isbn= 0-935702-11-3}} Chapter 16.
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| | |
| *{{cite book | first = Orazio | last = Svelto | title = Principles of Lasers | edition=5th | year=2010 }}
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| | |
| *{{cite book | first =Amnon | last =Yariv | year =1989 | title = Quantum Electronics| edition =3rd | publisher =Wiley | isbn =0-471-60997-8}}
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| | |
| *{{cite arxiv
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| | author=F. Pampaloni and J. Enderlein
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| | title=Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer
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| | journal=
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| | year = 2004
| |
| | eprint = physics/0410021
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| | class=physics.optics
| |
| }}
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| *{{cite journal
| |
| | author= G. Goubau and F. Schwering
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| | title=On the guided propagation of electromagnetic wave beams
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| | journal= IRE Trans.
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| | volume = 9
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| | year = 1961
| |
| | pages = 248-256
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| | doi= 10.1109/TAP.1961.1144999
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| | MR = 0134166 }}
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| *{{cite journal
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| | author= Miguel A. Bandres and Julio C. Gutierrez-Vega
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| | title=Ince Gaussian beams
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| | journal= Opt. Lett.
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| | pages = 144–146
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| | publisher = OSA
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| | volume = 29
| |
| | year = 2004
| |
| | url = http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-2-144
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| | doi= 10.1364/OL.29.000144
| |
| | pmid= 14743992
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| | issue= 2
| |
| |bibcode = 2004OptL...29..144B }}
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| *{{cite journal
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| | author= E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato
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| | title= Hypergeometric-Gaussian beams
| |
| | journal= Opt. Lett.
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| | pages = 3053–3055
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| | publisher = OSA
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| | volume = 32
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| | year = 2007
| |
| | url = http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-21-3053
| |
| | doi= 10.1364/OL.32.003053
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| | pmid= 17975594
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| | issue= 21
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| |bibcode = 2007OptL...32.3053K |arxiv = 0712.0782 }}
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| * [http://www.cvimellesgriot.com/Products/Documents/TechnicalGuide/Gaussian-Beam-Optics.pdf Gaussian Beam Propagation] - CVI Melles Griot Technical Guide
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| * [http://www.newport.com/Gaussian-Beam-Optics/144899/1033/content.aspx Gaussian Beam Optics Tutorial, Newport]
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| {{DEFAULTSORT:Gaussian Beam}}
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| [[Category:Laser science]]
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| [[Category:Electromagnetic radiation]]
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