orbital angular momentum of light

File:Helix oam.png

A beam of light carries a linear momentum $\backslash mathbf\{P\}$, and hence it can be also attributed an external angular momentum $\backslash mathbf\{L\}\_e=\backslash mathbf\{r\}\backslash times\backslash mathbf\{P\}$. 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 and depends on the spatial distribution of the optical field (E).

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). If the phase varies around the axis of such a wave, it carries orbital angular momentum.{{cite magazine |first=Alan E. |last=Willner |date=4 August 2016 |magazine=IEEE Spectrum |title=Twisted light could dramatically boost data rates: Orbital angular momentum could take optical and radio communication to new heights |url=https://spectrum.ieee.org/twisted-light-could-dramatically-boost-data-rates}}

In the figure to the right, the first column shows the beam wavefront shape. The second column is the 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).

The helical modes are characterized by an integer number $m$, positive or negative. If $m=0$, 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 $m=\backslash pm\; 1$, the handedness determined by the sign of $m$, the wavefront is shaped as a single helical surface, with a step length equal to the wavelength $\backslash lambda$. If $|m|\backslash geqslant\; 2$, the wavefront is composed of $|m|$ distinct but intertwined helices, with the step length of each helix surface equal to $|m|\backslash lambda$, and a handedness given by the sign of $m$. The integer $m$ is also the so-called "topological charge" of the optical vortex. Light beams that are in a helical mode carry nonzero OAM. As an example, any Laguerre-Gaussian mode with rotational mode number $l\; \backslash ne\; 0$ has such a helical wavefront.{{cite book|last=Siegman|first=Anthony E.|title=Lasers|url=https://archive.org/details/lasers0000sieg|url-access=registration|year=1986|publisher=University Science Books|isbn=978-0-935702-11-8|pages=[https://archive.org/details/lasers0000sieg/page/1283 1283]}}

The classical expression of the orbital angular momentum is the following:{{cite journal |last=Belinfante|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=449–474 |doi=10.1016/S0031-8914(40)90091-X |issue=5 |bibcode = 1940Phy.....7..449B |citeseerx=10.1.1.205.8093}}

$$\backslash mathbf\{L\}\; =\; \backslash epsilon\_0\; \backslash sum\_\{i=x,y,z\}\; \backslash int\; \backslash left(E^i\backslash left(\backslash mathbf\{r\}\; \backslash times\; \backslash boldsymbol\{\backslash nabla\}\backslash right)\; A^i\backslash right)d^3\backslash mathbf\{r\}\; ,$$

where $\backslash mathbf\{E\}$ and $\backslash mathbf\{A\}$ are the electric field and the vector potential, respectively, $\backslash epsilon\_0$ is the vacuum permittivity and we are using SI units. The $i$-superscripted symbols denote the cartesian components of the corresponding vectors.

For a monochromatic wave this expression can be transformed into the following one:{{cite journal|last=Humblet| first=J.| title=Sur le moment d'impulsion d'une onde electromagnetique| journal=Physica | year=1943| volume=10 |pages=585–603 |doi=10.1016/S0031-8914(43)90626-3 |issue=7 |bibcode = 1943Phy....10..585H }}{{citation |journal= IEEE Journal on Multiscale and Multiphysics Computational Techniques |last1=Sha| first1= Wei E. I.| last2= Lan| first2= Zhihao| last3=Chen| first3= Menglin L. N. |last4= Chen| first4= Yongpin P.|last5= Sun| first5= Sheng | title= Spin and Orbital Angular Momenta of Electromagnetic Waves: From Classical to Quantum Forms| year=2024| volume=9|pages=113–117|doi=10.1109/JMMCT.2024.3370729 |arxiv= 2403.01504|bibcode=2024IJMMC...9..113S }}

$$\backslash mathbf\{L\}=\backslash frac\{\backslash epsilon\_0\}\{2i\backslash omega\}\backslash sum\_\{i=x,y,z\}\; \backslash int\; \backslash left(\{E^i\}^\{\backslash ast\}\backslash left(\backslash mathbf\{r\}\backslash times\backslash mathbf\{\backslash nabla\}\backslash right)\; E^i\backslash right)\; d^3\backslash mathbf\{r\}\; .$$

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:

$$\backslash mathbf\{L\}\_z\; =\; m\backslash hbar\; .$$

where the topological charge $m$ can be extracted numerically from electric field profile of vortex beams. {{citation |title= OAM-Propagation| url=https://github.com/Sha-Group/OAM-Propagation}}

The corresponding wave functions (eigenfunctions of OAM operator) have the following general expression:

$$\backslash langle\; \backslash mathbf\{r\}|m\backslash rangle\backslash propto\; e^\{i\; m\; \backslash phi\}\; .$$

where $\backslash phi$ 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.

File:Generation of OAM beams using SLM.gif

Orbital angular momentum states with $l\; =\; \backslash pm\; 1$ occur naturally.{{Citation needed|date=November 2021}} OAM states of arbitrary $l$ can be created artificially using a variety of tools, such as using spiral phase plates, spatial light modulators and q-plates.

Spiral wave plates, made of plastic or glass, are plates where the thickness of the material increases in a spiral pattern in order to imprint a phase gradient on light passing through it. For a given wavelength, an OAM state of a given $l$ requires that the step height —the height between the thinnest and thickest parts of the plate— be given by $s\; =\; l\backslash lambda/(n-1)$ where $n$ is the refractive index of the plate. Although the wave plates themselves are efficient, they are relatively expensive to produce, and are, in general, not adjustable to different wavelengths of light.{{cite journal |last1 = Beijersbergen |first1 = M.W. |last2 = Coerwinkel |first2 = R.P.C. |last3 = Kristensen |first3 = M. |last4 = Woerdman |first4 = J.P. |author-link4=Han Woerdman |title = Helical-wavefront laser beams produced with a spiral phaseplate |journal = Optics Communications |date = December 1994 |volume = 112 |issue = 5–6 |pages = 321–327 |doi = 10.1016/0030-4018(94)90638-6 |bibcode = 1994OptCo.112..321B }}

Another way to modify the phase of the light is with a diffraction grating. For an $l=0$ state, the diffraction grating would consist of parallel lines. However, for an $l=1$ state, there will be a "fork" dislocation, and the number of lines above the dislocation will be one larger than below. An OAM state with $l>1$ can be created by increasing the difference in the number of lines above and below the dislocation.{{cite journal |last1 = Bazhenov |first1 = V.Yu. |last2 = Soskin |first2 = M.S. |last3 = Vasnetsov |first3 = M.V. |title = Screw Dislocations in Light Wavefronts |journal = Journal of Modern Optics |date = May 1992 |volume = 39 |issue = 5 |pages = 985–990 |doi = 10.1080/09500349214551011 |bibcode = 1992JMOp...39..985B }} As with the spiral wave plates, these diffraction gratings are fixed for $l$, but are not restricted to a particular wavelength.

A spatial light modulator operates in a similar way to diffraction gratings, but can be controlled by computer to dynamically generate a wide range of OAM states.

{{technical|section|date=July 2017}}

Theoretical work suggests that a series of optically distinct chromophores are capable of supporting an excitonic state whose symmetry is such that in the course of the exciton relaxing, a radiation mode of non-zero topological charge is created directly.{{cite journal |last1 = Williams |first1 = M.D. |title = Direct generation of optical vortices |journal = Physical Review A |date = March 2014 |volume = 89 |pages = 033837 |doi = 10.1103/PhysRevA.89.033837 |issue = 3 |last2 = Coles |first2 = M.M. |last3 = Bradshaw |first3 = D.S. |last4 = Andrews |first4 = D.L. |bibcode = 2014PhRvA..89c3837W |url = https://ueaeprints.uea.ac.uk/56744/1/PRA_twisted.pdf }}

Most recently,{{Cite journal |last1=Marrucci |first1=L. |last2=Manzo |first2=C. |last3=Paparo |first3=D. |date=2006-04-28 |title=Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media |url=https://link.aps.org/doi/10.1103/PhysRevLett.96.163905 |journal=Physical Review Letters |language=en |volume=96 |issue=16 |page=163905 |doi=10.1103/PhysRevLett.96.163905 |pmid=16712234 |issn=0031-9007|arxiv=0712.0099 |bibcode=2006PhRvL..96p3905M }} the geometric phase concept has been adopted for OAM generation. The geometric phase is modulated to coincide with the spatial phase dependence factor, i.e., $e^\{i\; m\; \backslash phi\}$ of an OAM carrying wave. In this way, geometric phase is introduced by using anisotropic scatterers.{{Cite journal |last1=Petrov |first1=Nikolay V. |last2=Sokolenko |first2=Bogdan |last3=Kulya |first3=Maksim S. |last4=Gorodetsky |first4=Andrei |last5=Chernykh |first5=Aleksey V. |date=2022 |title=Design of broadband terahertz vector and vortex beams: I. Review of materials and components |url=https://doi.org/10.37188/lam.2022.043 |journal=Light: Advanced Manufacturing |volume=3 |issue=4 |pages=1 |doi=10.37188/lam.2022.043 |issn=2831-4093}}{{Cite journal |last1=Petrov |first1=Nikolay V. |last2=Sokolenko |first2=Bogdan |last3=Kulya |first3=Maksim S. |last4=Gorodetsky |first4=Andrei |last5=Chernykh |first5=Aleksey V. |date=2022 |title=Design of broadband terahertz vector and vortex beams: II. Holographic assessment |url=https://doi.org/10.37188/lam.2022.044 |journal=Light: Advanced Manufacturing |volume=3 |issue=4 |pages=1 |doi=10.37188/lam.2022.044 |issn=2831-4093}} For example, a metamaterial composed of distributed linear polarizers in a rotational symmetric manner generates an OAM of order 1.{{cite journal |doi = 10.1364/JOSAB.29.000572 |title = Twisted Vector Field from an Inhomogeneous and Anisotropic Metamaterial |date = 2012-03-06 |last1 = Kang |first1 = Ming |last2 = Chen |first2 = Jing |last3 = Wang |first3 = Xi-Lin |last4 = Wang |first4 = Hui-Tian |journal = Journal of the Optical Society of America B |volume = 29 |issue = 4 |pages = 572–576 |bibcode = 2012JOSAB..29..572K }} To generate higher-order OAM wave, nano-antennas which can produce the spin-orbit coupling effect are designed and then arranged to form a metasurface with different topological charges.{{cite journal |doi = 10.1063/1.4895620 |title = Optical Spin-to-Orbital Angular Momentum Conversion in Ultra-Thin Metasurfaces with Arbitrary Topological Charges |date = 2014-09-11 |last1 = Bouchard |first1 = Frederic |last2 = Leon |first2 = Israel De |last3 = Schulz |first3 = Sebastian A. |last4 = Upham |first4 = Jeremy |last5 = Karimi |first5 = Ebrahim |last6 = Boyd |first6 = Robert W. |journal = Appl. Phys. Lett. |volume = 105 |issue = 10 |pages = 101905 |arxiv = 1407.5491 |bibcode = 2014ApPhL.105j1905B |s2cid = 39733399 }} Consequently, the transmitted wave carries an OAM, and its order is twice the value of the topological charge. Usually, the conversion efficiency is not high for the transmission-type metasurface. Alternative solution to achieve high transmittance is to use complementary (Babinet-inverted) metasurface.{{cite journal |doi = 10.1109/TAP.2016.2626722 |title = Ultrathin Complementary Metasurface for Orbital Angular Momentum Generation at Microwave Frequencies |date = 2016-11-08 |last1 = Chen |first1 = Menglin L. N. |last2 = Jiang |first2 = Li Jun |last3 = Sha |first3 = Wei E. I. |journal = IEEE Trans. Antennas Propag. |volume = 65 |issue = 1 |pages = 396–400 |arxiv = 1611.02814 |bibcode = 2017ITAP...65..396C |s2cid = 8222925 }} On the other hand, it is much easier to achieve high conversion efficiency, even 100% efficiency in the reflection-type metasurface such as the composite PEC-PMC metasurface.{{cite journal |doi = 10.1063/1.4941696 |title = Artificial Perfect Electric Conductor-Perfect Magnetic Conductor Anisotropic Metasurface for Generating Orbital Angular Momentum of Microwave with Nearly Perfect Conversion Efficiency |date = 2016-02-11 |last1 = Chen |first1 = Menglin L. N. |last2 = Jiang |first2 = Li Jun |last3 = Sha |first3 = Wei E. I. |journal = J. Appl. Phys. |volume = 119 |issue = 6 |pages = 064506 |arxiv = 1602.04557 |bibcode = 2016JAP...119f4506C |s2cid = 119208338 }}

Beside OAM generation in free space, integrated photonic approaches can also realize on-chip optical vortices carrying OAM. Representative approaches include patterned ring resonators,{{Cite journal |last1=Cai |first1=Xinlun |last2=Wang |first2=Jianwei |last3=Strain |first3=Michael J. |last4=Johnson-Morris |first4=Benjamin |last5=Zhu |first5=Jiangbo |last6=Sorel |first6=Marc |last7=O’Brien |first7=Jeremy L. |last8=Thompson |first8=Mark G. |last9=Yu |first9=Siyuan |date=2012-10-19 |title=Integrated Compact Optical Vortex Beam Emitters |url=https://www.science.org/doi/10.1126/science.1226528 |journal=Science |language=en |volume=338 |issue=6105 |pages=363–366 |doi=10.1126/science.1226528 |pmid=23087243 |bibcode=2012Sci...338..363C |s2cid=206543391 |issn=0036-8075}} subwavelength holographic gratings,{{Cite journal |last1=Zhou |first1=Nan |last2=Zheng |first2=Shuang |last3=Cao |first3=Xiaoping |last4=Zhao |first4=Yifan |last5=Gao |first5=Shengqian |last6=Zhu |first6=Yuntao |last7=He |first7=Mingbo |last8=Cai |first8=Xinlun |last9=Wang |first9=Jian |date=2019-05-03 |title=Ultra-compact broadband polarization diversity orbital angular momentum generator with 3.6 × 3.6 μm 2 footprint |journal=Science Advances |language=en |volume=5 |issue=5 |pages=eaau9593 |doi=10.1126/sciadv.aau9593 |issn=2375-2548 |pmc=6544453 |pmid=31172022|bibcode=2019SciA....5.9593Z }} Non-Hermitian vortex lasers,{{Cite journal |last1=Zhang |first1=Zhifeng |last2=Zhao |first2=Haoqi |last3=Pires |first3=Danilo Gomes |last4=Qiao |first4=Xingdu |last5=Gao |first5=Zihe |last6=Jornet |first6=Josep M. |last7=Longhi |first7=Stefano |last8=Litchinitser |first8=Natalia M. |last9=Feng |first9=Liang |date=2020-10-21 |title=Ultrafast control of fractional orbital angular momentum of microlaser emissions |journal=Light: Science & Applications |language=en |volume=9 |issue=1 |pages=179 |doi=10.1038/s41377-020-00415-3 |pmid=33101659 |pmc=7576132 |bibcode=2020LSA.....9..179Z |issn=2047-7538}}{{Cite journal |last1=Zhao |first1=Han |last2=Qiao |first2=Xingdu |last3=Wu |first3=Tianwei |last4=Midya |first4=Bikashkali |last5=Longhi |first5=Stefano |last6=Feng |first6=Liang |date=2019-09-13 |title=Non-Hermitian topological light steering |journal=Science |language=en |volume=365 |issue=6458 |pages=1163–1166 |doi=10.1126/science.aay1064 |pmid=31515392 |bibcode=2019Sci...365.1163Z |s2cid=202566887 |issn=0036-8075|doi-access=free }} and meta-waveguide OAM emitters.{{Cite journal |last1=He |first1=Tiantian |last2=Meng |first2=Yuan |last3=Liu |first3=Zhoutian |last4=Hu |first4=Futai |last5=Wang |first5=Rui |last6=Li |first6=Dan |last7=Yan |first7=Ping |last8=Liu |first8=Qiang |last9=Gong |first9=Mali |last10=Xiao |first10=Qirong |date=2021-11-22 |title=Guided mode meta-optics: metasurface-dressed waveguides for arbitrary mode couplers and on-chip OAM emitters with a configurable topological charge |url=https://opg.optica.org/abstract.cfm?URI=oe-29-24-39406 |journal=Optics Express |language=en |volume=29 |issue=24 |pages=39406–39418 |doi=10.1364/OE.443186 |pmid=34809306 |bibcode=2021OExpr..2939406H |s2cid=243813207 |issn=1094-4087|doi-access=free }}{{Cite journal |last1=Meng |first1=Yuan |last2=Chen |first2=Yizhen |last3=Lu |first3=Longhui |last4=Ding |first4=Yimin |last5=Cusano |first5=Andrea |last6=Fan |first6=Jonathan A. |last7=Hu |first7=Qiaomu |last8=Wang |first8=Kaiyuan |last9=Xie |first9=Zhenwei |last10=Liu |first10=Zhoutian |last11=Yang |first11=Yuanmu |last12=Liu |first12=Qiang |last13=Gong |first13=Mali |last14=Xiao |first14=Qirong |last15=Sun |first15=Shulin |date=2021-11-22 |title=Optical meta-waveguides for integrated photonics and beyond |journal=Light: Science & Applications |language=en |volume=10 |issue=1 |pages=235 |doi=10.1038/s41377-021-00655-x |pmid=34811345 |pmc=8608813 |bibcode=2021LSA....10..235M |issn=2047-7538}}

Determining the spin angular momentum (SAM) of light is simple – SAM is related to the polarization state of the light: the AM is, per photon, in a left and right circularly polarized beam respectively. Thus the SAM can be measured by transforming the circular polarization of light into a p- or s-polarized state by means of a wave plate and then using a polarizing beam splitter that will transmit or reflect the state of light.

The development of a simple and reliable method for the measurement of orbital angular momentum (OAM) of light, however, remains an important problem in the field of light manipulation. OAM (per photon) arises from the amplitude cross-section of the beam and is therefore independent of the spin angular momentum: whereas SAM has only two orthogonal states, the OAM is described by a state that can take any integer value N.{{cite book|last1=Padgett|first1=[ed.:] L. Allen, Stephen M. Barnett, Miles J.|title=Optical angular momentum|date=2003|publisher=Institute of Physics Publ.|location=Bristol [u.a.]|isbn=978-0-7503-0901-1}} As the state of OAM of light is unbounded, any integer value of l is orthogonal to (independent from) all the others. Where a beam splitter could separate the two states of SAM, no device can separate the N (if greater than 2) modes of OAM, and, clearly, the perfect detection of all N potential states is required to finally resolve the issue of measuring OAM. Nevertheless, some methods have been investigated for the measurement of OAM.

In practical scenarios, light beams carrying orbital angular momentum (OAM) are often not composed of a single pure mode due to distortions caused by atmospheric turbulence, optical misalignment, or scattering. These imperfections result in a superposition of multiple OAM modes within the beam. Measuring the OAM spectrum is essential to quantify the modal composition and assess the beam's quality, which is critical for applications such as optical communications, imaging, and quantum information processing.

To determine the vortex mode spectrum, one common method involves decomposing the complex optical field into its constituent angular harmonics:{{Cite journal |last1=Radivon |first1=Arina V. |last2=Katyba |first2=Gleb M. |last3=Raginov |first3=Nikita I. |last4=Chernykh |first4=Aleksey V. |last5=Ezerskii |first5=Aleksei S. |last6=Tsiplakova |first6=Elizaveta G. |last7=Rakov |first7=Ignat I. |last8=Paukov |first8=Maksim I. |last9=Starchenko |first9=Vladimir V. |last10=Arsenin |first10=Aleksey V. |last11=Spector |first11=Igor E. |last12=Zaytsev |first12=Kirill I. |last13=Krasnikov |first13=Dmitry V. |last14=Petrov |first14=Nikolay V. |last15=Nasibulin |first15=Albert G. |date=June 2024 |title=Expanding THz Vortex Generation Functionality with Advanced Spiral Zone Plates Based on Single-Walled Carbon Nanotube Films |url=https://onlinelibrary.wiley.com/doi/10.1002/adom.202303282 |journal=Advanced Optical Materials |language=en |volume=12 |issue=17 |doi=10.1002/adom.202303282 |issn=2195-1071}} For a scalar complex field expressed in polar coordinates, this decomposition is:

$u(\backslash rho,\; \backslash phi,\; z)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash sum\_\{l=-\backslash infty\}^\{\backslash infty\}\; a\_l(\backslash rho,\; z)\; \backslash exp(i\; l\; \backslash phi),$

where the summation runs over all integer OAM modes $l$. This decomposition requires knowledge of both the amplitude and phase of the field. While direct phase measurement is challenging, phase retrieval algorithms — particularly single-beam multiple-image reconstruction (SBMIR) methods{{Cite journal |last1=Petrov |first1=Nikolay V. |last2=Perraud |first2=Jean-Baptiste |last3=Chopard |first3=Adrien |last4=Guillet |first4=Jean-Paul |last5=Smolyanskaya |first5=Olga A. |last6=Mounaix |first6=Patrick |date=2020-08-01 |title=Terahertz phase retrieval imaging in reflection |url=https://opg.optica.org/ol/abstract.cfm?uri=ol-45-15-4168 |journal=Optics Letters |language=en |volume=45 |issue=15 |pages=4168–4171 |doi=10.1364/OL.397935 |pmid=32735250 |bibcode=2020OptL...45.4168P |issn=0146-9592}}{{Cite journal |last1=Tsiplakova |first1=Elizaveta G. |last2=Perraud |first2=Jean-Baptiste |last3=Guillet |first3=Jean-Paul |last4=Mounaix |first4=Patrick |last5=Petrov |first5=Nikolay V. |date=2023-11-01 |title=Terahertz diffractive imaging with saturated data inpainting |url=https://opg.optica.org/abstract.cfm?URI=ol-48-21-5463 |journal=Optics Letters |language=en |volume=48 |issue=21 |pages=5463–5466 |doi=10.1364/OL.499478 |pmid=37910678 |bibcode=2023OptL...48.5463T |issn=0146-9592}} — enable reliable recovery of the phase profile from intensity measurements alone, achieving high resolution without auxiliary reference beams.

The coefficients $a\_l(\backslash rho,\; z)$, representing the amplitude of each mode, are obtained by projecting the field onto the corresponding angular harmonic:

$a\_l(\backslash rho,\; z)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{0\}^\{2\backslash pi\}\; u(\backslash rho,\; \backslash phi,\; z)\; \backslash exp(-i\; l\; \backslash phi)d\backslash phi.$

The normalized intensity (or power) of each OAM mode $l$ at a propagation distance $z$ is then calculated by integrating the squared modulus of $a\_l$ over the radial coordinate $\backslash rho$:

$P\_l(z)\; =\; \backslash frac\{\backslash int\_\{0\}^\{\backslash infty\}\; |a\_l(\backslash rho,\; z)|^2\; \backslash rho\; d\backslash rho\}\{\backslash sum\_\{l\text{'}=-\backslash infty\}^\{\backslash infty\}\; \backslash int\_\{0\}^\{\backslash infty\}\; |a\_\{l\}(\backslash rho,\; z)|^2\; \backslash rho\; d\backslash rho\}.$

Here, the denominator ensures normalization by the total power across all modes, making $P\_l(z)$ a dimensionless fraction between 0 and 1. This spectrum provides a direct measure of the beam’s OAM purity and mode distribution.

Beams carrying OAM have a helical phase structure. Interfering such a beam with a uniform plane wave reveals phase information about the input beam through analysis of the observed spiral fringes. In a Mach–Zender interferometer, a helically phased source beam is made to interfere with a plane-wave reference beam along a collinear path. Interference fringes will be observed in the plane of the beam waist and/or at the Rayleigh range. The path being collinear, these fringes are pure consequence of the relative phase structure of the source beam. Each fringe in the pattern corresponds to one step through: counting the fringes suffices to determine the value of l.

Computer-generated holograms can be used to generate beams containing phase singularities, and these have now become a standard tool for the generation of beams carrying OAM. This generating method can be reversed: the hologram, coupled to a single-mode fiber of set entrance aperture, becomes a filter for OAM. This approach is widely used for the detection of OAM at the single-photon level.

The phase of these optical elements results to be the superposition of several fork-holograms carrying topological charges selected in the set of values to be demultiplexed. The position of the channels in far-field can be controlled by multiplying each fork-hologram contribution to the corresponding spatial frequency carrier.{{cite journal |last1=Ruffato |first1=Gianluca |last2=Massari |first2=Michele |last3=Romanato |first3=Filippo |title=Diffractive optics for combined spatial- and mode- division demultiplexing of optical vortices: design, fabrication and optical characterization |journal=Scientific Reports |date=20 April 2016 |volume=6 |issue=1 |pages=24760 |doi=10.1038/srep24760 |pmid=27094324 |pmc=4837364 |bibcode=2016NatSR...624760R |doi-access=free }}

Other methods to measure the OAM of light include the rotational Doppler effect, systems based on a Dove prism interferometer,{{cite journal |last1=Zhang |first1=Wuhong |last2=Qi |first2=Qianqian |last3=Zhou |first3=Jie |last4=Chen |first4=Lixiang |title=Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light |journal=Physical Review Letters |date=14 April 2014 |volume=112 |issue=15 |pages=153601 |doi=10.1103/PhysRevLett.112.153601 |pmid=24785038 |bibcode=2014PhRvL.112o3601Z }} the measure of the spin of trapped particles, the study of diffraction effects from apertures, and optical transformations.{{cite journal |last1=Berkhout |first1=Gregorius C. G. |last2=Lavery |first2=Martin P. J. |last3=Courtial |first3=Johannes |last4=Beijersbergen |first4=Marco W. |last5=Padgett |first5=Miles J. |title=Efficient Sorting of Orbital Angular Momentum States of Light |journal=Physical Review Letters |date=4 October 2010 |volume=105 |issue=15 |pages=153601 |doi=10.1103/PhysRevLett.105.153601 |pmid=21230900 |bibcode=2010PhRvL.105o3601B |hdl=1887/63550 |s2cid=3856640 |hdl-access=free }}{{cite journal |last1=Ruffato |first1=Gianluca |last2=Massari |first2=Michele |last3=Parisi |first3=Giuseppe |last4=Romanato |first4=Filippo |title=Test of mode-division multiplexing and demultiplexing in free-space with diffractive transformation optics |journal=Optics Express |date=3 April 2017 |volume=25 |issue=7 |pages=7859–7868 |doi=10.1364/OE.25.007859 |pmid=28380904 |arxiv=1612.06215 |bibcode=2017OExpr..25.7859R |s2cid=46850221 }} The latter use diffractive optical elements in order to unwrap the angular phase patterns of OAM modes into plane-wave phase patterns which can subsequently be resolved in the Fourier space. The resolution of such schemes can be improved by spiral transformations that extend the phase range of the output strip-shaped modes by the number of spirals in the input beamwidth.{{cite journal |last1=Wen |first1=Yuanhui |last2=Chremmos |first2=Ioannis |last3=Chen |first3=Yujie |last4=Zhu |first4=Jiangbo |last5=Zhang |first5=Yanfeng |last6=Yu |first6=Siyuan |title=Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes |journal=Physical Review Letters |date=11 May 2018 |volume=120 |issue=19 |pages=193904 |doi=10.1103/PhysRevLett.120.193904 |pmid=29799240 |arxiv=1801.08320 |bibcode=2018PhRvL.120s3904W |s2cid=44135155 }}

{{main|Orbital angular momentum multiplexing}}

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 DVDs or 320 gigabytes) per second.{{cite news |title='Twisted light' carries 2.5 terabits of data per second |url=https://www.bbc.co.uk/news/science-environment-18551284 |newspaper=BBC |date=25 June 2012 |access-date=25 June 2012}} Further research into OAM multiplexing in the radio and mm wavelength frequencies has been shown in preliminary tests to be able to transmit 32 gigabits of data per second over the air. The fundamental communication limit of orbital-angular-momentum multiplexing is increasingly urgent for current multiple-input multiple-output (MIMO) research. The limit has been clarified in terms of independent scattering channels or the degrees of freedom (DoF) of scattered fields through angular-spectral analysis, in conjunction with a rigorous Green function method.{{cite journal|last1= Yuan|first1= Shuai S. A.|last2= Wu|first2= Jie|last3= Chen|first3= Menglin L. N.|last4= Lan|first4= Zhihao|last5= Zhang|first5= Liang|last6= Sun|first6= Sheng|last7= Huang|first7= Zhixiang|last8= Chen|first8= Xiaoming|last9= Zheng|first9= Shilie|last10= Jiang|first10= Lijun|last11= Zhang|first11= Xianmin|last12= Sha|first12= Wei E. I.|title= Approaching the Fundamental Limit of Orbital-Angular-Momentum Multiplexing Through a Hologram Metasurface|journal=Physical Review Applied|date=16 December 2021|volume=16|issue= 6|doi=10.1103/PhysRevApplied.16.064042|pages= 064042|arxiv= 2106.15120|bibcode= 2021PhRvP..16f4042Y|s2cid= 245269914}}

The DoF limit is universal for arbitrary spatial-mode multiplexing, which is launched by a planar electromagnetic device, such as antenna, metasurface, etc., with a predefined physical aperture.

OAM states can be generated in coherent superpositions and they can be entangled,{{cite journal |last1=Joseph |first1=S.K. |last2=Chew |first2=L. Y. |first3=Sanjuan|last3=M.A. F.| title=Effect of geometry on the classical entanglement in a chaotic optical fiber |journal=Optics Express |date=10 Nov 2015 |volume=23 |issue=25 |pages=32191–32201 |doi=10.1364/OE.23.032191 |pmid=26699009 |bibcode=2015OExpr..2332191J |url=https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-23-25-32191&id=333200|doi-access=free }}{{Cite journal|last1=Pecoraro|first1=A.|last2=Cardano|first2=F.|last3=Marrucci|first3=L.|last4=Porzio|first4=A.|date=2019-07-15|title=Continuous-variable entangled states of light carrying orbital angular momentum|url=https://link.aps.org/doi/10.1103/PhysRevA.100.012321|journal=Physical Review A|language=en|volume=100|issue=1|pages=012321|doi=10.1103/PhysRevA.100.012321|arxiv=1805.05105|bibcode=2019PhRvA.100a2321P|s2cid=73549820|issn=2469-9926}} which is an integral element of schemes for quantum information protocols. Photon pairs generated by the process of parametric down-conversion are naturally entangled in OAM,{{cite journal |last1=Mair |first1=A. |last2=Vaziri |first2=A. |last3=Weihs |first3=G. |last4=Zeilinger |first4=A. |title=Entanglement of the orbital angular momentum states of photons |journal=Nature |date=2001 |volume=412 |issue=6844 |pages=313–316 |doi=10.1038/35085529|pmid=11460157 |arxiv=quant-ph/0104070 |bibcode=2001Natur.412..313M |s2cid=4401328 }}{{cite journal |last1=Walborn|first1=S.P. |last2=Oliveira |first2=A.N. |last3=Thebaldi |first3=R.S. |last4=Monken |first4=C.H. |title=Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion |journal=Physical Review A |date=2004 |volume=69 |issue=2 |pages=023811 |doi=10.1103/PhysRevA.69.023811|arxiv=quant-ph/0503029 |bibcode=2004PhRvA..69b3811W |s2cid=119490103 }} and correlations measured using spatial light modulators (SLM).{{cite journal |last1=Jack |first1=B. |last2=Yao |first2=A. M. |last3=Leach |first3=J. |last4=Romero |first4=J. |last5=Franke-Arnold |first5=S.|author5-link=Sonja Franke-Arnold |last6=Ireland |first6=D. G. |last7=Barnett |first7=S. M. |last8=Padgett |first8=M. J. |title=Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces |journal=Physical Review A |date=30 April 2010 |volume=81 |issue=4 |pages=043844 |doi=10.1103/PhysRevA.81.043844 |bibcode=2010PhRvA..81d3844J |url=https://espace.library.uq.edu.au/view/UQ:409754/UQ409754_OA.pdf }}

Using qudits (with d levels, as opposed to a qubit's 2 levels) has been shown to improve the robustness of quantum key distribution schemes. OAM states provide a suitable physical realisation of such a system, and a proof-of-principle experiment (with 7 OAM modes from $l\; =\; -3$ to $l\; =\; 3$) has been demonstrated.{{cite journal |last1=Mirhosseini |first1=Mohammad |last2=Magaña-Loaiza |first2=Omar S. |last3=O'Sullivan |first3=Malcolm N. |last4=Rodenburg |first4=Brandon |last5=Malik |first5=Mehul |last6=Lavery |first6=Martin P. J. |last7=Padgett |first7=Miles J. |last8=Gauthier |first8=Daniel J. |last9=Boyd |first9=Robert W. |title=High-dimensional quantum cryptography with twisted light |journal=New Journal of Physics |date=20 March 2015 |volume=17 |issue=3 |pages=033033 |doi=10.1088/1367-2630/17/3/033033 |arxiv=1402.7113 |bibcode=2015NJPh...17c3033M |s2cid=5300819 }}

In 2019, a letter published in the Monthly Notices of the Royal Astronomical Society presented evidence that OAM radio signals had been received from the vicinity of the M87* black hole, over 50 million light years distant, suggesting that optical angular momentum information can propagate over astronomical distances.{{cite journal |last1=Tamburini |first1=Fabrizio |last2=Thidé |first2=Bo |last3=Della Valle |first3=Massimo |title=Measurement of the spin of the M87 black hole from its observed twisted light |journal=Monthly Notices of the Royal Astronomical Society: Letters |date=February 2020 |volume=492 |issue=1 |pages=L22–L27 |doi=10.1093/mnrasl/slz176 |doi-access=free |arxiv=1904.07923 |bibcode=2020MNRAS.492L..22T}}

{{div col|colwidth=22em}}

• Angular momentum
• Angular momentum of light
• Orbital angular momentum of free electrons
• Circular polarization
• Hypergeometric-Gaussian modes
• Laguerre-Gaussian modes
• Spin angular momentum of light
• Paraxial approximation
• Polarization (waves)
• [https://patents.google.com/patent/US20160292472 Siae Microelettronica patent]

{{div col end}}

{{Reflist}}

{{Commons cat|Orbital angular momentum of waves}}

• [https://web.archive.org/web/20131217221347/http://www.phorbitech.eu/ Phorbitech]

{{refbegin}}

• {{Cite book |last1=Allen |first1=L. |name-list-style=amp |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}}.
• {{Cite book |last1= Torres |first1=Juan P. |name-list-style=amp |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}}.
• {{cite book|last1=Andrews|first1=David L.|name-list-style=amp |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/}}

{{refend}}

• [https://web.archive.org/web/20110927142704/http://www.physics.gla.ac.uk/Optics/ Glasgow Optics Group]
• [https://web.archive.org/web/20110921001433/http://molphys.leidenuniv.nl/qo/ Leiden Institute of Physics]
• [http://www.icfo.es/ ICFO]
• [https://web.archive.org/web/20051214055534if_/http://people.na.infn.it:80/~marrucci/softmattergroup/ Università Di Napoli "Federico II" (Archived copy)]
• [http://quantumoptics.phys.uniroma1.it/homepage.htm Università Di Roma "La Sapienza"]
• [http://sqogroup.ca University of Ottawa]
• [http://spookyactionbook.com/2016/04/10/how-to-demonstrate-twisted-light-using-a-laser-pointer/ Elementary demonstration using a laser pointer]

Category:Light

Category:Angular momentum of light