formulasearchengine - User contributions [en]

Circular buffer

2015-01-08T12:40:16Z

109.186.102.99: Last sentence stated: "if two elements are now removed then what would be returned is '''not''' 3 & 4 but A & B" which was incorrect (not FIFO) and didn't correspond with the image.

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Margin classifier

2014-10-18T09:39:58Z

109.186.126.84: Typo: is-->if

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Weber number

2013-12-30T23:21:50Z

109.186.34.9:

{{About|physical systems|the property of an infinite word|Critical exponent of a word}}
'''Critical exponents''' describe the behaviour of physical quantities near continuous [[phase transitions]]. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on
* the dimension of the system,
* the range of the interaction,
* the [[Spin (physics)|spin]] dimension.
These properties of critical exponents are supported by experimental data. The experimental results can be theoretically achieved in [[mean field theory]] for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires the [[renormalization group]].
Phase transitions and critical exponents appear also in percolation systems.

==Definition==
[[Phase transitions]] occur at a certain temperature, called the [[critical temperature]] <math>T_c</math>. We want to describe the behaviour of a physical quantity <math>f</math> in terms of a [[power law]] around the critical temperature. So we introduce the [[reduced temperature]] <math>\tau := (T-T_c)/T_c</math>, which is zero at the [[phase transition]], and define the critical exponent <math>k</math>.

:<math>k \, \stackrel{\text{def}}{=} \, \lim_{\tau \to 0}{\log |f(\tau)| \over \log |\tau|} \text{.}</math>

This results in the power law we were looking for.
:<math> f(\tau) \propto \tau^k, \quad \tau\approx 0 \text{.}</math>

It is important to remember that this represents the asymptotic behavior of the function <math>f(\tau)</math> as <math>\tau \to 0</math>.

More generally one might expect
:<math>f(\tau)=A \tau^k (1+b\tau ^{k_1} + \cdots) \text{.}</math>

==The most important critical exponents==
Below <math>T_c</math> the system has two different phases characterized by an [[order parameter]] <math>\Psi</math>, which vanishes at and above <math>T_c</math>.

Let us consider the [[disordered phase]] (<math>\tau</math> > 0), [[ordered phase]] (<math>\tau</math> < 0 ) and [[critical temperature]] (<math>\tau</math> = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It's also another standard convention to use the super/subscript +(-) for the disordered(ordered) state. We have [[spontaneous symmetry breaking]] in the ordered phase. So, we will arbitrarily take any solution in the phase.
{| style="border:#ddd 1px solid"
! colspan=2 | Keys
|-
| <math>\Psi</math>
| order parameter (e.g. <math> \frac{\rho-\rho_c}{\rho_c} </math> for the liquid-gas critical point, [[magnetization]] for the [[Curie point]],etc.)
|-
| <math>\tau</math>
| <math>\frac{T-T_c}{T_c}</math>
|-
| <math>f</math>
| specific [[Thermodynamic free energy|free energy]]
|-
| <math>C</math>
| [[specific heat]]; <math>-T\frac{\partial^2 f}{\partial T^2}</math>
|-
| <math>J</math>
| source field (e.g. <math>\frac{P-P_c}{P_c}</math> where ''P'' is the [[pressure]] and ''P''c the [[critical pressure]] for the liquid-gas critical point, reduced [[chemical potential]], the [[magnetic field]] ''H'' for the [[Curie point]] )
|-
| <math>\chi</math>
| the [[susceptibility]]/[[compressibility]]/etc.; <math>\frac{\partial \Psi}{\partial J}</math>
|-
| <math>\xi</math>
| [[correlation length]]
|-
| <math>d</math>
| the number of spatial [[dimension]]s
|-
| <math>\left\langle \psi(\vec{x}) \psi(\vec{y}) \right\rangle</math>
| the [[correlation function (statistical mechanics)|correlation function]]
|-
| <math>r</math>
| spatial distance
|}

The following entries are evaluated at <math>J = 0</math> (except for the <math>\delta</math> entry)
{| style="border:#ddd 1px solid"
|-
! colspan=2 | Critical exponents for <math>\tau</math> > 0 (disordered phase)
|-
| Greek letter
| relation
|-
| <math>\alpha</math>
| <math>C \propto \tau^{-\alpha}</math>
|-
| <math>\gamma</math>
| <math>\chi \propto \tau^{-\gamma}</math>
|-
| <math>\nu</math>
| <math>\xi \propto \tau^{-\nu}</math>
|-
|-
! colspan=2 | Critical exponents for <math>\tau</math> < 0 (ordered phase)
|-
| Greek letter
| relation
|-
| <math>\alpha^\prime</math>
| <math>C \propto (-\tau)^{-\alpha^\prime}</math>
|-
| <math>\beta</math>
| <math>\Psi \propto (-\tau)^{\beta}</math>
|-
| <math>\gamma^\prime</math>
| <math>\chi \propto (-\tau)^{-\gamma^\prime}</math>
|-
| <math>\nu^\prime</math>
| <math>\xi \propto (-\tau)^{-\nu^\prime}</math>
|-
|-
! Critical exponents for <math>\tau</math> = 0
|-
| Greek letter
| relation
|-
| <math>\delta</math>
| <math>J \propto \Psi^\delta</math>
|-
| <math>\eta</math>
| <math>\left\langle \psi(0) \psi(r) \right\rangle \propto r^{-d+2-\eta}</math>
|}

The critical exponents can be derived from the specific free energy <math>f(J,T)</math> as a function of the source and temperature. The correlation length can be derived from the [[functional (mathematics)|functional]] <math>F[J;T]</math>.

These relations are accurate close to the critical point in two- and three-dimensional systems. In four dimensions, however, the power laws are modified by logarithmic factors.
This problem does not appear in [[dimensional regularization|3.99 dimensions]], though. 

== Mean field critical exponents of Ising-like systems ==

The classical [[Landau theory]] (aka [[mean field theory]]) values of the critical exponents for a scalar field (of which the Ising-model is the prototype example) are given by

:<math>\alpha = \alpha^\prime = 0\,</math>
:<math>\beta = \frac{1}{2}\,</math>
:<math>\gamma = \gamma^\prime = 1\,</math>
:<math>\delta = 3\,</math>

If we add derivative terms turning it into a mean field [[Ginzburg–Landau theory]], we get

:<math>\eta = 0\,</math>

:<math>\nu = \frac{1}{2}\,</math>

One of the major discoveries in the study of critical phenomena is that mean field theory of critical points is only correct when the space dimension of the system is four or higher (which unfortunately excludes many of the experimentally relevant cases). This dimension is called the upper critical dimension. The problem with mean field theory is that the critical exponents do not depend on the space dimension. This leads to a quantitative discrepancy in space dimensions 2 and 3, where the true critical exponents differ from the mean field values. It leads to a qualitative discrepancy in space dimension 1, where a critical point in fact no longer exists, even though mean field theory still predicts there is one. The space dimension where mean field theory becomes qualitatively incorrect is called the lower critical dimension.

== Experimental values ==

The most accurately measured value of <math>\alpha</math> is −0.0127 for the phase transition of superfluid helium (the so-called [[lambda transition]]). The value was measured in a satellite to minimize pressure differences in the sample ([http://xxx.arxiv.org/abs/cond-mat/0310163 see here]). This result agrees with theoretical prediction obtained by [[variational perturbation theory]] ([http://prola.aps.org/abstract/PRD/v60/i8/e085001 see here] or [http://www.physik.fu-berlin.de/~kleinert/kleiner_re279/seventh.html here]).

== Scaling functions ==
In light of the critical scalings, we can reexpress all thermodynamic quantities in terms of dimensionless quantities. Close enough to the critical point, everything can be reexpressed in terms of certain ratios of the powers of the reduced quantities. These are the scaling functions.

The origin of scaling functions can be seen from the renormalization group. The critical point is an [[infrared fixed point]]. In a sufficiently small neighborhood of the critical point, we may linearize the action of the renormalization group. This basically means that rescaling the system by a factor of ''a'' will be equivalent to rescaling operators and source fields by a factor of <math>a^\Delta</math> for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.

== Scaling relations ==

:<math>\alpha \equiv \alpha'</math>
:<math>\gamma \equiv \gamma'</math>
:<math>\nu \equiv \nu'</math>

Thus, the exponents above and below the critical temperature, respectively, have identical values. This is understandable, since the respective scaling functions, <math> f_\pm(k\xi ,\dots)</math>, originally defined for <math>k\xi \ll 1</math>, should become identical if extrapolated to <math> k\xi \gg 1\,.</math>

Critical exponents are denoted by Greek letters. They fall into [[universality classes]] and obey the [[scaling relation]]s

:<math> \nu d = 2 - \alpha = 2\beta + \gamma = \beta(\delta + 1) = \gamma \frac{\delta + 1}{\delta - 1}\,</math>

:<math> 2 - \eta = \frac{\gamma}{\nu} = d \frac{\delta - 1}{\delta + 1}</math>

These equations imply that there are only two independent exponents, e.g., <math>\,\nu</math> and {{nowrap|<math>\eta\,</math>.}} All this follows from the theory of the [[renormalization group]].

== Anisotropy ==

There are some [[anisotropic]] systems where the correlation length is direction dependent.

== Multicritical points ==
More complex behaviour may occur at [[multicritical point]]s, at the border or on intersections of critical manifolds.

==Static versus dynamic properties==
The above examples exclusively refer to the static properties of a critical system. However dynamic properties of the system may become critical, too. Especially, the characteristic time, <math>\tau_{\mathrm{char}}</math>, of a system diverges as <math>\tau_{\mathrm{char}}\propto \xi^z</math>, with a ''dynamical exponent'' <math>z</math>. Moreover, the large ''static universality classes'' of equivalent models with identical static critical exponents decompose into smaller ''dynamical universality classes'', if one demands that also the dynamical exponents are identical.

The critical exponents can be computed from [[conformal field theory]].

See also [[anomalous scaling dimension]].

== Transport properties ==
Critical exponents also exist for transport quantities like [[viscosity]] and [[heat conductivity]].

== Self-organized criticality ==
Critical exponents also exist for self organized criticality for [[dissipative system]]s.

==Percolation Theory==
Phase transitions and critical exponents appear also in percolation processes where the concentration of occupied sites or links play the role of temperature. See [[percolation critical exponents]].

== See also ==
* [[Rushbrooke inequality]]
* [[Widom scaling]]
* [[Ising critical exponents]]

==External links and literature==
* [[Hagen Kleinert]] and Verena Schulte-Frohlinde, ''[http://www.physik.fu-berlin.de/~kleinert/b8 Critical Properties of φ4-Theories]'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4658-7
* Toda, M., Kubo, R., N. Saito, ''Statistical Physics I'', Springer-Verlag (Berlin, 1983); Hardcover ISBN 3-540-11460-2
* J.M.Yeomans, ''Statistical Mechanics of Phase Transitions'', Oxford Clarendon Press
* [[H. Eugene Stanley|H. E. Stanley]] ''Introduction to Phase Transitions and Critical Phenomena'', Oxford University Press, 1971
* A. Bunde and [[Shlomo Havlin|S. Havlin]] (editors), ''[http://havlin.biu.ac.il/Shlomo%20Havlin%20books_f_in_s.php Fractals in Science]'', Springer, 1995
* A. Bunde and [[Shlomo Havlin|S. Havlin]] (editors), ''[http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php Fractals and Disordered Systems]'', Springer, 1996

[[Category:Phase transitions]]
[[Category:Critical phenomena]]
[[Category:Renormalization group]]

Fresnel zone

2013-12-01T07:59:39Z

109.186.33.233: Fixed typo

[[Image:Gaussian beam w40mm lambda30mm.png|thumb|right|Instantaneous intensity of a Gaussian beam.]]
[[Image:Green laser pointer TEM00 profile.JPG|thumb|right|A 5 mW green laser pointer beam profile, showing the TEM00 profile]]
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 "TEM00 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.

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.

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>

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.
[[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.]]

==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" />

:<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>

where<ref name="svelto153" />
:<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"),
:<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''2 of their axial values, respectively,
:<math>w_0 = w(0)</math> is the [[#Beam width or spot size|waist size]],
:<math>R(z)</math> is the [[#Radius of curvature|radius of curvature]] of the beam's wavefronts, and
:<math>\zeta(z)</math> is the [[#Gouy phase|Gouy phase shift]], an extra contribution to the phase that is seen in Gaussian beams.
Additionally, the field has a time dependence factor <math>e^{i\omega t}</math> that has been suppressed in the above expression.

The corresponding time-averaged [[intensity (physics)|intensity]] (or [[irradiance]]) distribution is

:<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>

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>.

== Beam parameters ==

The geometry and behavior of a Gaussian beam are governed by a set of '''beam parameters''', which are defined in the following sections.

===Beam width or spot size===
{{see also|Beam diameter}}
[[Image:GaussianBeamWaist.svg|thumb|350px|right|Gaussian beam width ''w''(''z'') as a function of the axial distance ''z''. ''w''0: beam waist; ''b'': depth of focus; ''z''R: [[Rayleigh range]]; <math>\Theta</math>: total angular spread]]

For a Gaussian beam propagating in free space, the spot size (radius) ''w''(''z'') will be at a minimum value ''w''0 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" />

:<math>w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } \ . </math>

where the origin of the z-axis is defined, without loss of generality, to coincide with the beam waist, and where<ref name="svelto153" />

:<math>z_\mathrm{R} = \frac{\pi w_0^2}{\lambda}</math>

is called the [[Rayleigh range]].

===Rayleigh range and confocal parameter===
At a distance from the waist equal to the Rayleigh range ''z''R, the width ''w'' of the beam is<ref name="svelto153" />

:<math> w(\pm z_\mathrm{R}) = \sqrt{2} w_0.</math>

The distance between these two points is called the ''confocal parameter'' or ''depth of focus'' of the beam:

:<math>b = 2 z_\mathrm{R} = \frac{2 \pi w_0^2}{\lambda}\,.</math>

=== Radius of curvature ===

''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" />

:<math>R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right] \ . </math>

===Beam divergence===
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" />

:<math>\theta \simeq \frac{\lambda}{\pi w_0} \qquad (\theta \mathrm{\ in\ radians}). </math>

The total angular spread of the beam far from the waist is then given by
:<math>\Theta = 2 \theta\ .</math>

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.

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>.

[[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''2 ("[[M squared]]"). The ''M''2 for a Gaussian beam is one. All real laser beams have ''M''2 values greater than one, although very high quality beams can have values very close to one.

===Gouy phase===
The ''longitudinal phase delay'' or ''Gouy phase'' of the beam is<ref name="svelto153" />

:<math>\zeta(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right) \ .</math>

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" />

===Complex beam parameter===
{{main|Complex beam parameter}}
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>

:<math> q(z) = z + q_0 = z + iz_\mathrm{R} \ .</math>

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" />

:<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>

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]].

In terms of the complex beam parameter <math>{q}</math>, a Gaussian field with one transverse dimension is proportional to

:<math>
{u}(x,z) = \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right).
</math>

In two dimensions one can write the potentially elliptical or astigmatic beam as the product

:<math>
{u}(x,y,z) = {u}(x,z)\, {u}(y,z),
</math>

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>

:<math>
{u}(r,z) = \frac{1}{{q}(z)}\exp\left( -i k\frac{r^2}{2 {q}(z)}\right).
</math>

==Power and intensity==

=== Power through an aperture ===
The [[power (physics)|power]] ''P'' passing through a circle of radius ''r'' in the transverse plane at position ''z'' is

:<math> P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]\ ,</math>

where

:<math> P_0 = { 1 \over 2 } \pi I_0 w_0^2 </math>

is the total power transmitted by the beam.

For a circle of radius <math>r = w(z) \, </math>, the fraction of power transmitted through the circle is

:<math>{ P(z) \over P_0 } = 1 - e^{-2} \approx 0.865\ .</math>

Similarly, about 95 percent of the beam's power will flow through a circle of radius <math>r = 1.224\cdot w(z) \, </math>.

=== Peak and average intensity ===
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>:

:<math>I(0,z) =\lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2}
= \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>

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>.

==Derivation==
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>
:<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" />
:<math> U(x,y,z,t) = u(x,y,z) e^{-i(kz-\omega t)},</math>
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" />
:<math>\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.</math>

Solving this differential equation yields an infinite set of functions, of which the Gaussian beam is the lowest-order solution or ''[[transverse mode|mode]]''.

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

=== Hermite-Gaussian modes === 
[[Image:Hermite-gaussian.png|thumb|250px|right|Twelve Hermite-Gaussian modes]]
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
:<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.

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>

Hermite-Gaussian modes are typically designated "TEM''mn''", where ''m'' and ''n'' are the polynomial indices in the x and y directions. A Gaussian beam is thus TEM00.

=== Laguerre-Gaussian modes === 
[[Image:LG-wiki.jpg|thumb|right|upright=1.5|The intensity profiles of twelve Laguerre-Gaussian modes]]
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
:<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],
</math>

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]].

=== 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>Bandres and Gutierrez-Vega (2004)</ref>

:<math>
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) \psi _{GS}\left( z\right) \right] ,
</math>
where <math>\xi</math> and <math>\eta</math> are the radial and angular elliptic coordinates
defined by

:<math>
x = \sqrt{\varepsilon /2}w\left( z\right) \cosh \xi \cos \eta ,
</math>
:<math>
y = \sqrt{\varepsilon /2}w\left( z\right) \sinh \xi \sin \eta.
</math>
<math>{C}_{p}^{m}\left( \eta ,\epsilon \right)</math> are the even Ince
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>
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.

=== Hypergeometric-Gaussian modes ===

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

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>

:<math>
u_{pm}(\rho,\theta;\zeta)=
\sqrt{\frac{2^{p+|m|+1}}{\pi\Gamma(p+|m|+1)}} \frac{\Gamma(1+|m|+\frac{p}{2})}{\Gamma(|m|+1)}
\,\,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),
</math>

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
function.

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.

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:

:<math>
u(\rho,\phi,0) \propto \rho^{p+|m|}e^{-\rho^2+im\phi}.
</math>

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]].

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

==Notes==
<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.

*{{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.

*{{cite book | first = Anthony E.|last=Siegman|year=1986|title=Lasers|publisher=University Science Books|isbn= 0-935702-11-3}} Chapter 16.

*{{cite book | first = Orazio | last = Svelto | title = Principles of Lasers | edition=5th | year=2010 }}

*{{cite book | first =Amnon | last =Yariv | year =1989 | title = Quantum Electronics| edition =3rd | publisher =Wiley | isbn =0-471-60997-8}}

*{{cite arxiv
| author=F. Pampaloni and J. Enderlein
| title=Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer
| journal=
| year = 2004
| eprint = physics/0410021
| class=physics.optics
}}
*{{cite journal
| author= G. Goubau and F. Schwering
| title=On the guided propagation of electromagnetic wave beams
| journal= IRE Trans.
| volume = 9
| year = 1961
| pages = 248-256
| doi= 10.1109/TAP.1961.1144999
| MR = 0134166 }}
*{{cite journal
| author= Miguel A. Bandres and Julio C. Gutierrez-Vega
| title=Ince Gaussian beams
| journal= Opt. Lett.
| pages = 144–146
| publisher = OSA
| volume = 29
| year = 2004
| url = http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-2-144
| doi= 10.1364/OL.29.000144
| pmid= 14743992
| issue= 2
|bibcode = 2004OptL...29..144B }}
*{{cite journal
| author= E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato
| title= Hypergeometric-Gaussian beams
| journal= Opt. Lett.
| pages = 3053–3055
| publisher = OSA
| volume = 32
| year = 2007
| url = http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-21-3053
| doi= 10.1364/OL.32.003053
| pmid= 17975594
| issue= 21
|bibcode = 2007OptL...32.3053K |arxiv = 0712.0782 }}
* [http://www.cvimellesgriot.com/Products/Documents/TechnicalGuide/Gaussian-Beam-Optics.pdf Gaussian Beam Propagation] - CVI Melles Griot Technical Guide
* [http://www.newport.com/Gaussian-Beam-Optics/144899/1033/content.aspx Gaussian Beam Optics Tutorial, Newport]

{{DEFAULTSORT:Gaussian Beam}}
[[Category:Laser science]]
[[Category:Electromagnetic radiation]]

Orders of magnitude (data)

2012-07-27T00:16:52Z

109.186.102.170: added date reference

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