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| The '''orbital angular momentum of light''' (OAM) is the component of [[Angular momentum of light|angular momentum]] of a light beam that is dependent on the field spatial distribution, and not on the [[Polarization (waves)|polarization]]. It can be further split into an internal and an external OAM. The internal OAM is an origin-independent angular momentum of a light beam that can be associated with a [[helical]] or twisted [[wavefront]]. The external OAM is the origin-dependent angular momentum that can be obtained as [[cross product]] of the light beam position (center of the beam) and its total [[linear momentum]].
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| == Introduction ==
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| [[File:Helix oam.png|thumb|right|400px|Different columns show the beam helical structures, phase fronts, and corresponding intensity distributions.]]
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| A beam of light carries a [[linear momentum]] <math>\mathbf{P}</math>, and hence it can be also attributed an external angular momentum <math>\mathbf{L}_e=\mathbf{r}\times\mathbf{P}</math>. This external angular momentum depends on the choice of the origin of the [[coordinate]] system. If one chooses the origin at the beam axis and the beam is cylindrically symmetric (at least in its momentum distribution), the external angular momentum will vanish. The external angular momentum is a form of OAM, because it is unrelated to [[Polarization (waves)|polarization]] and depends on the spatial distribution of the [[optical field]].
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| A more interesting example of OAM is the internal OAM appearing when a [[paraxial]] light beam is in a so-called “''helical mode''”. Helical modes of the [[electromagnetic field]] are characterized by a [[wavefront]] that is shaped as a [[helix]], with an [[optical vortex]] in the center, at the beam axis (see figure). The helical modes are characterized by an integer number <math>m</math>, positive or negative. If <math>m=0</math>, the mode is not helical and the wavefronts are multiple disconnected surfaces, for example, a sequence of parallel planes (from which the name “plane wave”). If <math>m=\pm 1</math>, the handedness determined by the sign of <math>m</math>, the [[wavefront]] is shaped as a single helical surface, with a step length equal to the [[wavelength]] <math>\lambda</math>. If <math>|m|\geqslant 2</math>, the wavefront is composed of <math>|m|</math> distinct but intertwined helices, with the step length of each helix surface equal to <math>|m|\lambda</math>, and a handedness given by the sign of <math>m</math>. The integer <math>m</math> is also the so-called “''topological charge''” of the [[optical vortex]]. Light beams that are in a helical mode carry nonzero OAM.
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| In the figure to the right, the first column shows the beam wavefront shape. The second column is the [[Optical phase space|optical phase]] distribution in a beam cross-section, shown in false colors. The third column is the light [[intensity]] distribution in a beam cross-section (with a dark vortex core at the center).
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| The beam photons in this case have an OAM of <math>m\hbar</math> directed along the beam axis. This OAM is origin-independent.
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| An example of [[optical modes]] having a helical [[wavefront]] is provided by the set of [[Laguerre-Gaussian]] modes.<ref>{{cite book|last=Siegmam|first=Anthony E.|title=Lasers|year=1986|publisher=University Science Books|isbn=0-935702-11-3|pages=1283}}</ref>
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| == Mathematical expressions for the orbital angular momentum of light ==
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| The classical expression of the orbital angular momentum in the paraxial limit is the following:<ref>{{cite journal|last=Belintante|first=F. J.|title=On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields|journal=Physica|year=1940|volume=7|pages=449|doi=10.1016/S0031-8914(40)90091-X|issue=5|bibcode = 1940Phy.....7..449B }}</ref>
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| :<math>\mathbf{L}=\epsilon_0\sum_{i=x,y,z}\int \left(E^i\left(\mathbf{r}\times\mathbf{\nabla}\right)A^i\right)d^{3}\mathbf{r} ,</math>
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| where <math>\mathbf{E}</math> and <math>\mathbf{A}</math> are the [[electric field]] and the [[vector potential]], respectively, <math>\epsilon_0</math> is the [[vacuum permittivity]] and we are using SI units. The <math>i</math>-superscripted symbols denote the cartesian components of the corresponding vectors.
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| For a monochromatic wave this expression can be transformed into the following one:<ref>{{cite journal|last=Humblet|first=J.|title=Sur le moment d'impulsion d'une onde electromagnetique|journal=Physica (Utrecht)|year=1943|volume=10|pages=585|doi=10.1016/S0031-8914(43)90626-3|issue=7|bibcode = 1943Phy....10..585H }}</ref>
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| :<math>\mathbf{L}=\frac{\epsilon_0}{2i\omega}\sum_{i=x,y,z}\int \left({E^i}^{\ast}\left(\mathbf{r}\times\mathbf{\nabla}\right)E^{i}\right)d^{3}\mathbf{r} .</math>
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| This expression is generally nonvanishing when the wave is not cylindrically symmetric. In particular, in a quantum theory, individual photons may have the following values of the OAM:
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| :<math>\mathbf{L}_z=m\hbar .</math>
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| The corresponding wave functions (eigenfunctions of OAM operator) have the following general expression:
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| :<math>\langle \mathbf{r}|m\rangle\propto e^{i m \phi} .</math>
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| where <math>\phi</math> is the cylindrical coordinate. As mentioned in the Introduction, this expression corresponds to waves having a helical wavefront (see figure above), with an optical vortex in the center, at the beam axis.
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| ==Potential use in telecommunications==
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| {{main|Orbital angular momentum multiplexing}}
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| Research into OAM has suggested that light waves could carry hitherto unprecedented quantities of data through [[optical fibres]]. According to preliminary tests, data streams travelling along a beam of light split into 8 different circular polarities have demonstrated the capacity to transfer up to 2.5 terabits of data (equivalent to 66 [[DVD]]s or 320 [[Gigabyte]]s) per second.<ref>{{cite news |title='Twisted light' carries 2.5 terabits of data per second |author= |url=http://www.bbc.co.uk/news/science-environment-18551284 |newspaper=BBC |date=25 June 2012 |accessdate=25 June 2012}}</ref>
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| ==See also==
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| <div style="-moz-column-count:3; -webkit-column-count:3; column-count:3;">
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| * [[Angular momentum]]
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| * [[Angular momentum of light]]
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| * [[Circular polarization]]
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| * [[Gaussian beam#Hypergeometric-Gaussian modes|Hypergeometric-Gaussian modes]]
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| * [[Laguerre-Gaussian modes]]
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| * [[Light spin angular momentum]]
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| * [[Optical vortices]]
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| * [[Paraxial approximation]]
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| * [[Polarization (waves)]]
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| </div>
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| == References ==
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| <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| == External links ==
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| * [http://www.phorbitech.eu/ Phorbitech]
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| * [http://www.physics.gla.ac.uk/Optics/ Glasgow Optics Group]
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| * [http://www.molphys.leidenuniv.nl/qo/ Leiden Institute of Physics]
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| * [http://www.icfo.es/ ICFO]
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| * [http://people.na.infn.it/~marrucci/softmattergroup/ Università Di Napoli "Federico II"]
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| * [http://quantumoptics.phys.uniroma1.it/homepage.htm Università Di Roma "La Sapienza"]
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| ==Further reading==
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| *{{Cite book |last=Allen |first=L. |lastauthoramp=yes |last2=Barnett |first2=Stephen M.|last3=Padgett |first3=Miles J. |title=Optical Angular Momentum |publisher=Institute of Physics |location=Bristol |year=2003 |isbn=978-0-7503-0901-1|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}.
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| *{{Cite book |last= Torres |first=Juan P. |lastauthoramp=yes |last2=Torner |first2= Lluis|title=Twisted Photons: Applications of Light with Orbital Angular Momentum |publisher= Wiley-VCH |location=Bristol |year=2011 |isbn=978-3-527-40907-5|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}.
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| *{{cite book|last=Andrews|first=David L.|lastauthoramp=yes |last2=Babiker |first2= Mohamed|title=The Angular Momentum of Light|year=2012|publisher=Cambridge University Press|location=Cambridge|isbn=9781107006348|pages=448|url=http://www.cambridge.org/de/knowledge/isbn/item6687744/The%20Angular%20Momentum%20of%20Light/}}
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| <!--- Categories --->
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| [[Category:Articles created via the Article Wizard]]
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| [[Category:Light]]
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Science Technicians Lamar from Markham, really likes blogging, ganhando dinheiro na internet and rowing. Likes to visit new towns and spots like Monastery of the Hieronymites and Tower of Belém in Lisbon.
Here is my homepage :: como ganhar dinheiro na internet