1556:
1264:
1021:
1274:
693:
1621:
1650:
1032:
1709:
806:
1667:
491:
1551:{\displaystyle \mathbf {J} \approx {\frac {{\hat {z}}\epsilon _{0}}{2\omega }}\int \left(\left|E_{\text{L}}\right|^{2}-\left|E_{\text{R}}\right|^{2}\right)d^{3}\mathbf {r} +{\frac {{\hat {z}}\varepsilon _{0}}{2i\omega }}\int \sum _{i=x,y,z}\left({E^{i}}^{\ast }{\frac {\partial }{\partial \phi }}E^{i}\right)d^{3}\mathbf {r} .}
154:, which is a property of all objects in rotational motion. For example, a light beam can be rotating around its own axis while it propagates forward. Again, the existence of this angular momentum can be made evident by transferring it to small absorbing or scattering particles, which are thus subject to an optical torque.
1688:
1259:{\displaystyle \mathbf {J} ={\frac {\varepsilon _{0}}{2i\omega }}\int \left(\mathbf {E} ^{\ast }\times \mathbf {E} \right)d^{3}\mathbf {r} +{\frac {\varepsilon _{0}}{2i\omega }}\sum _{i=x,y,z}\int \left({E^{i}}^{\ast }\left(\mathbf {r} \times {\boldsymbol {\nabla }}\right)E^{i}\right)d^{3}\mathbf {r} .}
1740:
The applications of the spin angular momentum of light are undistinguishable from the innumerable applications of the light polarization and will not be discussed here. The possible applications of the orbital angular momentum of light are instead currently the subject of research. In particular, the
1640:
crystals are commonly used to manipulate the light polarization. Whenever the polarization ellipticity is changed, in the process, there is an exchange of SAM between light and the crystal. If the crystal is free to rotate, it will do so. Otherwise, the SAM is finally transferred to the holder and to
1628:
When a light beam carrying nonzero angular momentum impinges on an absorbing particle, its angular momentum can be transferred on the particle, thus setting it in rotational motion. This occurs both with SAM and OAM. However, if the particle is not at the beam center the two angular momenta will give
2692:
Boyd, R.W.; Jha, Anand; Malik, Mehul; O'Sullivan, Colin; Rodenburg, Brandon; Gauthier, Daniel J. (2011). Hasan, Zameer U; Hemmer, Philip R; Lee, Hwang; Santori, Charles M (eds.). "Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon".
728:
These two expressions can be proved to be equivalent to each other for any electromagnetic field that satisfies
Maxwell’s equations with no source charges and vanishes fast enough outside a finite region of space. The two terms in the second expression however are physically ambiguous, as they are
1268:
Let us now consider the paraxial limit, with the beam axis assumed to coincide with the z axis of the coordinate system. In this limit the only significant component of the angular momentum is the z one, that is the angular momentum measuring the light beam rotation around its own axis, while the
1629:
rise to different kinds of rotation of the particle. SAM will give rise to a rotation of the particle around its own center, i.e., to a particle spinning. OAM, instead, will generate a revolution of the particle around the beam axis. These phenomena are schematically illustrated in the figure.
176:, namely SAM and OAM. However this distinction becomes blurred for strongly focused or diverging beams, and in the general case only the total angular momentum of a light field can be defined. An important limiting case in which the distinction is instead clear and unambiguous is that of a "
1025:
A justification for taking this step is yet to be provided. The latter expression has further problems, as it can be shown that the two terms are not true angular momenta as they do not obey the correct quantum commutation rules. Their sum, that is the total angular momentum, instead does.
1016:{\displaystyle \mathbf {J} _{\perp }=\varepsilon _{0}\int \left(\mathbf {E} _{\perp }\times \mathbf {A} _{\perp }\right)d^{3}\mathbf {r} +\varepsilon _{0}\sum _{i=x,y}\int \left({E^{i}}_{\perp }\left(\mathbf {r} \times {\boldsymbol {\nabla }}\right)A_{\perp }^{i}\right)d^{3}\mathbf {r} .}
352:
688:{\displaystyle \mathbf {J} =\varepsilon _{0}\int \left(\mathbf {E} \times \mathbf {A} \right)d^{3}\mathbf {r} +\varepsilon _{0}\sum _{i=x,y,z}\int \left(E^{i}\left(\mathbf {r} \times {\boldsymbol {\nabla }}\right)A^{i}\right)d^{3}\mathbf {r} =\mathbf {S} +\mathbf {L} ,}
2630:
Malik, M.; O’Sullivan, Malcolm; Rodenburg, Brandon; Mirhosseini, Mohammad; Leach, Jonathan; Lavery, Martin P. J.; Padgett, Miles J.; Boyd, Robert W. (2012). "Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding".
1657:
In the paraxial limit, the OAM of a light beam can be exchanged with material media that have a transverse spatial inhomogeneity. For example, a light beam can acquire OAM by crossing a spiral phase plate, with an inhomogeneous thickness (see figure).
17:
270:
1699:
is a device, currently realized using liquid crystals, polymers or sub-wavelength gratings, which can generate OAM by exploiting a SAM sign-change. In this case, the OAM sign is controlled by the input polarization.
1674:
A more convenient approach for generating OAM is based on using diffraction on a fork-like or pitchfork hologram (see figure). Holograms can be also generated dynamically under the control of a computer by using a
30:
inserted into the beam. The spin component causes the particle to spin around its axis, while the other component, known as orbital angular momentum (OAM), causes the particle to rotate around the axis of the
2399:
Karimi, E.; Piccirillo, Bruno; Nagali, Eleonora; Marrucci, Lorenzo; Santamato, Enrico (2009). "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates".
21:
20:
22:
801:
772:
172:
fields around the propagation direction, and the second with the dynamical rotation of light rays around the main beam axis. These two rotations are associated with two forms of
423:
1610:
1583:
715:
486:
460:
396:
374:
1695:
Another method for generating OAM is based on the SAM-OAM coupling that may occur in a medium which is both anisotropic and inhomogeneous. In particular, the so-called
19:
1850:
Belinfante, F. J. (1940). "On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields".
2488:
Allen, L.; Beijersbergen, M.; Spreeuw, R.; Woerdman, J. (1992). "Orbital angular momentum of light and the transformation of
Laguerre-Gaussian laser modes".
1732:
placed at a specific distance (see figure) in order to introduce a well-defined relative phase between horizontal and vertical
Hermite-Gaussian beams.
1741:
following applications have been already demonstrated in research laboratories, although they have not yet reached the stage of commercialization:
234:, because it depends on the spatial distribution of the field. However, since its value is dependent from the choice of the origin, it is termed "
2453:
Gecevicius, M.; Drevinskas, R.; Beresna, M.; Kazansky, P.G. (2014). "Single beam optical vortex tweezers with tunable orbital angular momentum".
2041:
O'Neil, A. T.; MacVicar, I.; Allen, L.; Padgett, M. (2002). "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam".
2930:
2909:
2890:
1819:
1029:
An equivalent but simpler expression for a monochromatic wave of frequency ω, using the complex notation for the fields, is the following:
2581:
Gibson, G.; Courtial, Johannes; Padgett, Miles J.; Vasnetsov, Mikhail; Pas'Ko, Valeriy; Barnett, Stephen M.; Franke-Arnold, Sonja (2004).
18:
2285:
Heckenberg, N. R.; McDuff, R; Smith, CP; White, AG (1992). "Generation of optical phase singularities by computer-generated holograms".
2949:
2338:
Marrucci, L.; Manzo, C.; Paparo, D. (2006). "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media".
2198:
Heckenberg, N. R.; McDuff, R.; Smith, C. P.; Rubinsztein-Dunlop, H.; Wegener, M. J. (1992). "Laser beams with phase singularities".
2124:
463:
241:
1632:
In the case of transparent media, in the paraxial limit, the optical SAM is mainly exchanged with anisotropic systems, for example
347:{\displaystyle \mathbf {J} =\varepsilon _{0}\int \mathbf {r} \times \left(\mathbf {E} \times \mathbf {B} \right)d^{3}\mathbf {r} ,}
2854:
1752:
2920:
2200:
1804:
437:
95:
1725:
1932:"Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity"
143:
phenomenon, in which a light beam transfers its momentum to an absorbing or scattering object, generating a mechanical
1809:
89:
2836:
26:
The total angular momentum of light consists of two components, both of which act in a different way on a massive
219:
In addition to these two terms, if the origin of coordinates is located outside the beam axis, there is a third
2841:
1676:
777:
748:
2302:
2005:
1867:
433:
1784:
1717:
264:
209:
131:
103:
47:
2831:
2794:
2702:
2650:
2594:
2547:
2497:
2462:
2419:
2357:
2294:
2251:
2172:
2097:
2050:
1997:
1946:
1904:
1859:
1824:
1789:
1758:
205:
119:
76:
2307:
2010:
1872:
1829:
1762:
734:
426:
401:
267:
is the following one, in which there is no explicit distinction between the two forms of rotation:
1588:
1561:
698:
469:
443:
379:
357:
2767:
2749:
2718:
2674:
2640:
2532:
2435:
2409:
2381:
2347:
2267:
2217:
2163:
Bazhenov, V.Yu.; Soskin, M.S.; Vasnetsov, M.V. (1992). "Screw
Dislocations in Light Wavefronts".
1794:
136:
27:
2785:
Foo, G.; Palacios, David M.; Swartzlander, Grover A. Jr. (2005). "Optical Vortex
Coronagraph".
2926:
2905:
2886:
2810:
2666:
2612:
2563:
2513:
2373:
2320:
2066:
2023:
1962:
1931:
1721:
54:. While traveling approximately in a straight line, a beam of light can also be rotating (or "
2802:
2759:
2710:
2658:
2602:
2555:
2505:
2470:
2427:
2365:
2312:
2259:
2236:
2209:
2180:
2105:
2058:
2015:
1954:
1912:
1877:
1814:
1779:
1746:
1729:
718:
260:
220:
190:
186:
173:
151:
84:
43:
1679:. As a result, this allows one to obtain arbitrary values of the orbital angular momentum.
2858:
2139:
2798:
2706:
2654:
2598:
2551:
2501:
2466:
2423:
2361:
2298:
2255:
2176:
2101:
2054:
2001:
1950:
1908:
1863:
436:
is the following one, in which there are two separate terms that may be associated with
2954:
83:
shape. These two forms of rotation are therefore associated with two distinct forms of
2851:
1916:
1881:
2943:
2863:
2771:
2583:"Free-space information transfer using light beams carrying orbital angular momentum"
2221:
2109:
1670:
Schematic showing generation of orbital angular momentum of light in a
Gaussian beam.
725:-superscripted symbols denote the cartesian components of the corresponding vectors.
194:
2740:(2008). "Beating the channel capacity limit for linear photonic superdense coding".
2722:
2678:
2439:
2385:
2271:
2085:
1666:
1637:
1633:
1620:
730:
2369:
2237:"Topological charge and angular momentum of light beams carrying optical vortices"
2062:
1649:
212:. OAM is related with the spatial field distribution, and in particular with the
1958:
1653:
Schematic of generating light orbital angular momentum with spiral phase plate.
2737:
2184:
1895:
Humblet, J. (1943). "Sur le moment d'impulsion d'une onde electromagnetique".
737:. A gauge-invariant version can be obtained by replacing the vector potential
223:
contribution obtained as the cross-product of the beam position and its total
2559:
2509:
2263:
2697:. Advances in Photonics of Quantum Computing, Memory, and Communication IV.
2088:(1994). "Helical-wavefront laser beams produced with a spiral phase plate".
1980:
Simpson, N. B.; Dholakia, K.; Allen, L.; Padgett, M. J. (1997). "Mechanical
1708:
213:
198:
80:
75:
There are two distinct forms of rotation of a light beam, one involving its
2814:
2670:
2616:
2607:
2582:
2377:
2324:
2235:
Soskin, M.; Gorshkov, V.; Vasnetsov, M.; Malos, J.; Heckenberg, N. (1997).
2070:
2027:
1966:
432:
However, another expression of the angular momentum naturally arising from
2567:
2531:
Friese, M. E. J.; Enger, J; Rubinsztein-Dunlop, H; Heckenberg, NR (1996).
2517:
1612:
denote the left and right circular polarization components, respectively.
2806:
2662:
2316:
2019:
1712:
pi/2-cylindrical mode converter transforms HG mode into a proper LG mode.
224:
178:
169:
165:
144:
127:
46:
quantity that expresses the amount of dynamical rotation present in the
2213:
1696:
2714:
2474:
2431:
68:) around its own axis. This rotation, while not visible to the naked
2763:
1757:
Higher-dimensional quantum information encoding, for possible future
123:
2902:
Twisted
Photons: Applications of Light with Orbital Angular Momentum
72:, can be revealed by the interaction of the light beam with matter.
1930:
He, H.; Friese, M.; Heckenberg, N.; Rubinsztein-Dunlop, H. (1995).
1687:
2754:
2645:
2533:"Optical angular-momentum transfer to trapped absorbing particles"
2414:
2352:
1799:
1745:
Orientational manipulation of particles or particle aggregates in
1707:
1691:
The q-plate effect for left and right-hand circular polarizations.
1686:
1665:
1648:
107:
51:
134:
motion. The existence of this momentum becomes apparent in the "
102:
The total angular momentum of light (or, more generally, of the
2881:
Allen, L.; Barnett, Stephen M. & Padgett, Miles J. (2003).
1736:
Possible applications of the orbital angular momentum of light
69:
2084:
Beijersbergen, M. W.; Coerwinkel, R.P.C.; Kristensen, M.;
255:
Mathematical expressions for the angular momentum of light
204:
For such a beam, SAM is strictly related with the optical
164:, the first associated with the dynamical rotation of the
2125:"Laser beams with screw dislocations in their wavefronts"
1624:
Spin and orbital angular momentum interaction with matter
1616:
Exchange of spin and orbital angular momentum with matter
2123:
Bazhenov, V.Yu.; Vasnetsov, M.V.; Soskin, M.S. (1990).
130:, which is a characteristic property of all objects in
2925:. Cambridge: Cambridge University Press. p. 448.
2868:
2846:
189:
beam in which all light rays (or, more precisely, all
1591:
1564:
1277:
1035:
809:
780:
751:
701:
494:
472:
446:
404:
382:
360:
273:
398:
are the electric and magnetic fields, respectively,
157:
For a light beam, one can usually distinguish two "
1604:
1577:
1550:
1258:
1015:
795:
766:
709:
687:
480:
454:
417:
390:
368:
346:
2919:Andrews, David L. & Babiker, Mohamed (2012).
745:with their “transverse” or radiative component
8:
2900:Torres, Juan P. & Torner, Lluis (2011).
1984:spin and orbital angular momentum of light:
803:, thus obtaining the following expression:
259:One commonly used expression for the total
1716:OAM can also be generated by converting a
2753:
2644:
2606:
2413:
2351:
2306:
2009:
1871:
1596:
1590:
1569:
1563:
1540:
1534:
1519:
1500:
1494:
1487:
1482:
1452:
1425:
1410:
1409:
1406:
1398:
1392:
1377:
1367:
1349:
1339:
1305:
1290:
1289:
1286:
1278:
1276:
1248:
1242:
1227:
1213:
1205:
1194:
1187:
1182:
1149:
1126:
1120:
1112:
1106:
1092:
1083:
1078:
1050:
1044:
1036:
1034:
1005:
999:
984:
979:
965:
957:
946:
939:
934:
907:
897:
885:
879:
864:
859:
849:
844:
829:
816:
811:
808:
787:
782:
779:
758:
753:
750:
702:
700:
677:
669:
661:
655:
640:
626:
618:
607:
571:
561:
549:
543:
529:
521:
507:
495:
493:
473:
471:
447:
445:
409:
403:
383:
381:
361:
359:
336:
330:
316:
308:
295:
286:
274:
272:
110:fields) and matter is conserved in time.
1619:
15:
1842:
1751:High-bandwidth information encoding in
208:, and in particular with the so-called
1820:Orbital angular momentum multiplexing
1269:other two components are negligible.
796:{\displaystyle \mathbf {E} _{\perp }}
767:{\displaystyle \mathbf {A} _{\perp }}
7:
35:Physical quantity carried in photons
2852:UniversitĂ Di Napoli "Federico II"
1506:
1502:
227:. This third term is also called "
197:) only form small angles with the
14:
2885:. Bristol: Institute of Physics.
2736:Barreiro, J. T.; Wei, Tzu-Chieh;
251:OAM appearing for helical beams.
2864:UniversitĂ Di Roma "La Sapienza"
1753:free-space optical communication
1636:crystals. Indeed, thin slabs of
1541:
1399:
1279:
1249:
1214:
1206:
1113:
1093:
1079:
1037:
1006:
966:
958:
886:
860:
845:
812:
783:
754:
703:
678:
670:
662:
627:
619:
550:
530:
522:
496:
474:
448:
384:
362:
337:
317:
309:
296:
275:
2201:Optical and Quantum Electronics
721:of the magnetic field, and the
1805:Light orbital angular momentum
1415:
1295:
96:light orbital angular momentum
1:
2922:The Angular Momentum of Light
2370:10.1103/PhysRevLett.96.163905
2063:10.1103/PhysRevLett.88.053601
1917:10.1016/S0031-8914(43)90626-3
1882:10.1016/S0031-8914(40)90091-X
1728:system with two well-aligned
418:{\displaystyle \epsilon _{0}}
2138:(8): 429–431. Archived from
2110:10.1016/0030-4018(94)90638-6
1605:{\displaystyle E_{\text{R}}}
1578:{\displaystyle E_{\text{L}}}
710:{\displaystyle \mathbf {A} }
481:{\displaystyle \mathbf {L} }
455:{\displaystyle \mathbf {S} }
391:{\displaystyle \mathbf {B} }
369:{\displaystyle \mathbf {E} }
118:Light, or more generally an
2842:Leiden Institute of Physics
1810:Light spin angular momentum
1768:Sensitive optical detection
1704:Cylindrical mode converters
429:and we are using SI units.
185:light beam, that is a well
90:light spin angular momentum
2971:
1959:10.1103/PhysRevLett.75.826
2950:Angular momentum of light
2185:10.1080/09500349214551011
40:angular momentum of light
2883:Optical Angular Momentum
2560:10.1103/PhysRevA.54.1593
2510:10.1103/PhysRevA.45.8185
2264:10.1103/PhysRevA.56.4064
2165:Journal of Modern Optics
1645:Spiral phase plate (SPP)
242:orbital angular momentum
2455:Applied Physics Letters
2402:Applied Physics Letters
2340:Physical Review Letters
2043:Physical Review Letters
1939:Physical Review Letters
1677:spatial light modulator
741:and the electric field
2904:. Bristol: Wiley-VCH.
2608:10.1364/OPEX.12.005448
1713:
1692:
1671:
1654:
1625:
1606:
1579:
1552:
1260:
1017:
797:
768:
711:
689:
482:
456:
419:
392:
370:
348:
147:on it in the process.
32:
2090:Optics Communications
1785:Circular polarization
1711:
1690:
1669:
1652:
1623:
1607:
1580:
1553:
1261:
1018:
798:
769:
712:
690:
483:
457:
420:
393:
371:
349:
265:electromagnetic field
244:, as opposed to the "
210:circular polarization
150:Light may also carry
104:electromagnetic field
87:, respectively named
48:electromagnetic field
25:
2869:University of Ottawa
2837:Glasgow Optics Group
2807:10.1364/OL.30.003308
2663:10.1364/OE.20.013195
2317:10.1364/OL.17.000221
2020:10.1364/OL.22.000052
1825:Polarization (waves)
1790:Electromagnetic wave
1759:quantum cryptography
1589:
1562:
1275:
1033:
807:
778:
749:
699:
492:
470:
444:
402:
380:
358:
271:
120:electromagnetic wave
2799:2005OptL...30.3308F
2707:2011SPIE.7948E..0LB
2655:2012OExpr..2013195M
2599:2004OExpr..12.5448G
2552:1996PhRvA..54.1593F
2502:1992PhRvA..45.8185A
2467:2014ApPhL.104w1110G
2424:2009ApPhL..94w1124K
2362:2006PhRvL..96p3905M
2299:1992OptL...17..221H
2256:1997PhRvA..56.4064S
2177:1992JMOp...39..985B
2102:1994OptCo.112..321B
2055:2002PhRvL..88e3601O
2002:1997OptL...22...52S
1951:1995PhRvL..75..826H
1909:1943Phy....10..585H
1864:1940Phy.....7..449B
1830:Photon polarization
1763:quantum computation
1662:Pitch-fork hologram
989:
427:vacuum permittivity
122:, carries not only
2857:2016-03-04 at the
2214:10.1007/BF01588597
1795:Helmholtz equation
1730:cylindrical lenses
1714:
1693:
1672:
1655:
1626:
1602:
1575:
1548:
1475:
1256:
1172:
1013:
975:
924:
793:
764:
707:
685:
594:
478:
452:
415:
388:
366:
344:
193:components of the
137:radiation pressure
79:and the other its
33:
28:colloidal particle
2932:978-1-107-00634-8
2911:978-3-527-40907-5
2892:978-0-7503-0901-1
2715:10.1117/12.873491
2639:(12): 13195–200.
2496:(11): 8185–8189.
2475:10.1063/1.4882418
2432:10.1063/1.3154549
1722:Laguerre-Gaussian
1599:
1572:
1513:
1448:
1443:
1418:
1370:
1342:
1320:
1298:
1145:
1143:
1067:
903:
567:
434:Noether’s theorem
159:forms of rotation
23:
2962:
2936:
2915:
2896:
2819:
2818:
2782:
2776:
2775:
2764:10.1038/nphys919
2757:
2733:
2727:
2726:
2689:
2683:
2682:
2648:
2627:
2621:
2620:
2610:
2578:
2572:
2571:
2546:(2): 1593–1596.
2537:
2528:
2522:
2521:
2485:
2479:
2478:
2450:
2444:
2443:
2417:
2396:
2390:
2389:
2355:
2335:
2329:
2328:
2310:
2282:
2276:
2275:
2241:
2232:
2226:
2225:
2195:
2189:
2188:
2160:
2154:
2153:
2151:
2150:
2144:
2129:
2120:
2114:
2113:
2081:
2075:
2074:
2038:
2032:
2031:
2013:
1977:
1971:
1970:
1936:
1927:
1921:
1920:
1892:
1886:
1885:
1875:
1847:
1815:Optical vortices
1780:Angular momentum
1747:optical tweezers
1724:one by using an
1718:Hermite-Gaussian
1611:
1609:
1608:
1603:
1601:
1600:
1597:
1584:
1582:
1581:
1576:
1574:
1573:
1570:
1557:
1555:
1554:
1549:
1544:
1539:
1538:
1529:
1525:
1524:
1523:
1514:
1512:
1501:
1499:
1498:
1493:
1492:
1491:
1474:
1444:
1442:
1431:
1430:
1429:
1420:
1419:
1411:
1407:
1402:
1397:
1396:
1387:
1383:
1382:
1381:
1376:
1372:
1371:
1368:
1354:
1353:
1348:
1344:
1343:
1340:
1321:
1319:
1311:
1310:
1309:
1300:
1299:
1291:
1287:
1282:
1265:
1263:
1262:
1257:
1252:
1247:
1246:
1237:
1233:
1232:
1231:
1222:
1218:
1217:
1209:
1199:
1198:
1193:
1192:
1191:
1171:
1144:
1142:
1131:
1130:
1121:
1116:
1111:
1110:
1101:
1097:
1096:
1088:
1087:
1082:
1068:
1066:
1055:
1054:
1045:
1040:
1022:
1020:
1019:
1014:
1009:
1004:
1003:
994:
990:
988:
983:
974:
970:
969:
961:
951:
950:
945:
944:
943:
923:
902:
901:
889:
884:
883:
874:
870:
869:
868:
863:
854:
853:
848:
834:
833:
821:
820:
815:
802:
800:
799:
794:
792:
791:
786:
773:
771:
770:
765:
763:
762:
757:
719:vector potential
716:
714:
713:
708:
706:
694:
692:
691:
686:
681:
673:
665:
660:
659:
650:
646:
645:
644:
635:
631:
630:
622:
612:
611:
593:
566:
565:
553:
548:
547:
538:
534:
533:
525:
512:
511:
499:
487:
485:
484:
479:
477:
461:
459:
458:
453:
451:
424:
422:
421:
416:
414:
413:
397:
395:
394:
389:
387:
375:
373:
372:
367:
365:
353:
351:
350:
345:
340:
335:
334:
325:
321:
320:
312:
299:
291:
290:
278:
261:angular momentum
250:
240:
233:
221:angular momentum
184:
174:angular momentum
163:
152:angular momentum
142:
85:angular momentum
67:
60:
24:
2970:
2969:
2965:
2964:
2963:
2961:
2960:
2959:
2940:
2939:
2933:
2918:
2912:
2899:
2893:
2880:
2877:
2875:Further reading
2859:Wayback Machine
2828:
2823:
2822:
2793:(24): 3308–10.
2784:
2783:
2779:
2735:
2734:
2730:
2691:
2690:
2686:
2629:
2628:
2624:
2593:(22): 5448–56.
2580:
2579:
2575:
2535:
2530:
2529:
2525:
2487:
2486:
2482:
2452:
2451:
2447:
2398:
2397:
2393:
2337:
2336:
2332:
2308:10.1.1.472.1077
2284:
2283:
2279:
2239:
2234:
2233:
2229:
2197:
2196:
2192:
2162:
2161:
2157:
2148:
2146:
2142:
2127:
2122:
2121:
2117:
2083:
2082:
2078:
2040:
2039:
2035:
2011:10.1.1.205.5751
1979:
1978:
1974:
1934:
1929:
1928:
1924:
1894:
1893:
1889:
1873:10.1.1.205.8093
1849:
1848:
1844:
1839:
1834:
1775:
1738:
1706:
1685:
1664:
1647:
1618:
1592:
1587:
1586:
1565:
1560:
1559:
1530:
1515:
1505:
1483:
1481:
1480:
1476:
1432:
1421:
1408:
1388:
1363:
1359:
1358:
1335:
1331:
1330:
1329:
1325:
1312:
1301:
1288:
1273:
1272:
1238:
1223:
1204:
1200:
1183:
1181:
1180:
1176:
1132:
1122:
1102:
1077:
1076:
1072:
1056:
1046:
1031:
1030:
995:
956:
952:
935:
933:
932:
928:
893:
875:
858:
843:
842:
838:
825:
810:
805:
804:
781:
776:
775:
752:
747:
746:
697:
696:
651:
636:
617:
613:
603:
602:
598:
557:
539:
520:
516:
503:
490:
489:
468:
467:
442:
441:
405:
400:
399:
378:
377:
356:
355:
326:
307:
303:
282:
269:
268:
257:
248:
238:
231:
216:helical shape.
182:
161:
140:
116:
65:
58:
36:
16:
12:
11:
5:
2968:
2966:
2958:
2957:
2952:
2942:
2941:
2938:
2937:
2931:
2916:
2910:
2897:
2891:
2876:
2873:
2872:
2871:
2866:
2861:
2849:
2844:
2839:
2834:
2827:
2826:External links
2824:
2821:
2820:
2787:Optics Letters
2777:
2742:Nature Physics
2738:Kwiat, Paul G.
2728:
2684:
2633:Optics Express
2622:
2587:Optics Express
2573:
2523:
2480:
2461:(23): 231110.
2445:
2408:(23): 231124.
2391:
2346:(16): 163905.
2330:
2287:Optics Letters
2277:
2227:
2190:
2155:
2115:
2086:Woerdman, J.P.
2076:
2033:
1990:Optics Letters
1982:equivalence of
1972:
1945:(5): 826–829.
1922:
1887:
1841:
1840:
1838:
1835:
1833:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1776:
1774:
1771:
1770:
1769:
1766:
1755:
1749:
1737:
1734:
1705:
1702:
1684:
1681:
1663:
1660:
1646:
1643:
1617:
1614:
1595:
1568:
1547:
1543:
1537:
1533:
1528:
1522:
1518:
1511:
1508:
1504:
1497:
1490:
1486:
1479:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1451:
1447:
1441:
1438:
1435:
1428:
1424:
1417:
1414:
1405:
1401:
1395:
1391:
1386:
1380:
1375:
1366:
1362:
1357:
1352:
1347:
1338:
1334:
1328:
1324:
1318:
1315:
1308:
1304:
1297:
1294:
1285:
1281:
1255:
1251:
1245:
1241:
1236:
1230:
1226:
1221:
1216:
1212:
1208:
1203:
1197:
1190:
1186:
1179:
1175:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1148:
1141:
1138:
1135:
1129:
1125:
1119:
1115:
1109:
1105:
1100:
1095:
1091:
1086:
1081:
1075:
1071:
1065:
1062:
1059:
1053:
1049:
1043:
1039:
1012:
1008:
1002:
998:
993:
987:
982:
978:
973:
968:
964:
960:
955:
949:
942:
938:
931:
927:
922:
919:
916:
913:
910:
906:
900:
896:
892:
888:
882:
878:
873:
867:
862:
857:
852:
847:
841:
837:
832:
828:
824:
819:
814:
790:
785:
761:
756:
705:
684:
680:
676:
672:
668:
664:
658:
654:
649:
643:
639:
634:
629:
625:
621:
616:
610:
606:
601:
597:
592:
589:
586:
583:
580:
577:
574:
570:
564:
560:
556:
552:
546:
542:
537:
532:
528:
524:
519:
515:
510:
506:
502:
498:
476:
450:
412:
408:
386:
364:
343:
339:
333:
329:
324:
319:
315:
311:
306:
302:
298:
294:
289:
285:
281:
277:
256:
253:
115:
112:
106:and the other
34:
13:
10:
9:
6:
4:
3:
2:
2967:
2956:
2953:
2951:
2948:
2947:
2945:
2934:
2928:
2924:
2923:
2917:
2913:
2907:
2903:
2898:
2894:
2888:
2884:
2879:
2878:
2874:
2870:
2867:
2865:
2862:
2860:
2856:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2829:
2825:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2781:
2778:
2773:
2769:
2765:
2761:
2756:
2751:
2747:
2743:
2739:
2732:
2729:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2688:
2685:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2652:
2647:
2642:
2638:
2634:
2626:
2623:
2618:
2614:
2609:
2604:
2600:
2596:
2592:
2588:
2584:
2577:
2574:
2569:
2565:
2561:
2557:
2553:
2549:
2545:
2541:
2534:
2527:
2524:
2519:
2515:
2511:
2507:
2503:
2499:
2495:
2491:
2484:
2481:
2476:
2472:
2468:
2464:
2460:
2456:
2449:
2446:
2441:
2437:
2433:
2429:
2425:
2421:
2416:
2411:
2407:
2403:
2395:
2392:
2387:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2354:
2349:
2345:
2341:
2334:
2331:
2326:
2322:
2318:
2314:
2309:
2304:
2300:
2296:
2292:
2288:
2281:
2278:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2238:
2231:
2228:
2223:
2219:
2215:
2211:
2207:
2203:
2202:
2194:
2191:
2186:
2182:
2178:
2174:
2170:
2166:
2159:
2156:
2145:on 2017-12-15
2141:
2137:
2133:
2126:
2119:
2116:
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2080:
2077:
2072:
2068:
2064:
2060:
2056:
2052:
2049:(5): 053601.
2048:
2044:
2037:
2034:
2029:
2025:
2021:
2017:
2012:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1976:
1973:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1933:
1926:
1923:
1918:
1914:
1910:
1906:
1902:
1898:
1891:
1888:
1883:
1879:
1874:
1869:
1865:
1861:
1857:
1853:
1846:
1843:
1836:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1777:
1772:
1767:
1764:
1760:
1756:
1754:
1750:
1748:
1744:
1743:
1742:
1735:
1733:
1731:
1727:
1723:
1719:
1710:
1703:
1701:
1698:
1689:
1682:
1680:
1678:
1668:
1661:
1659:
1651:
1644:
1642:
1639:
1635:
1630:
1622:
1615:
1613:
1593:
1566:
1545:
1535:
1531:
1526:
1520:
1516:
1509:
1495:
1488:
1484:
1477:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1449:
1445:
1439:
1436:
1433:
1426:
1422:
1412:
1403:
1393:
1389:
1384:
1378:
1373:
1364:
1360:
1355:
1350:
1345:
1336:
1332:
1326:
1322:
1316:
1313:
1306:
1302:
1292:
1283:
1270:
1266:
1253:
1243:
1239:
1234:
1228:
1224:
1219:
1210:
1201:
1195:
1188:
1184:
1177:
1173:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1146:
1139:
1136:
1133:
1127:
1123:
1117:
1107:
1103:
1098:
1089:
1084:
1073:
1069:
1063:
1060:
1057:
1051:
1047:
1041:
1027:
1023:
1010:
1000:
996:
991:
985:
980:
976:
971:
962:
953:
947:
940:
936:
929:
925:
920:
917:
914:
911:
908:
904:
898:
894:
890:
880:
876:
871:
865:
855:
850:
839:
835:
830:
826:
822:
817:
788:
759:
744:
740:
736:
732:
726:
724:
720:
682:
674:
666:
656:
652:
647:
641:
637:
632:
623:
614:
608:
604:
599:
595:
590:
587:
584:
581:
578:
575:
572:
568:
562:
558:
554:
544:
540:
535:
526:
517:
513:
508:
504:
500:
465:
439:
435:
430:
428:
410:
406:
341:
331:
327:
322:
313:
304:
300:
292:
287:
283:
279:
266:
262:
254:
252:
247:
243:
237:
230:
226:
222:
217:
215:
211:
207:
202:
200:
196:
195:optical field
192:
188:
181:
180:
175:
171:
167:
160:
155:
153:
148:
146:
139:
138:
133:
132:translational
129:
125:
121:
113:
111:
109:
105:
100:
98:
97:
92:
91:
86:
82:
78:
73:
71:
64:
57:
53:
49:
45:
41:
29:
2921:
2901:
2882:
2790:
2786:
2780:
2745:
2741:
2731:
2698:
2694:
2687:
2636:
2632:
2625:
2590:
2586:
2576:
2543:
2540:Phys. Rev. A
2539:
2526:
2493:
2490:Phys. Rev. A
2489:
2483:
2458:
2454:
2448:
2405:
2401:
2394:
2343:
2339:
2333:
2290:
2286:
2280:
2247:
2244:Phys. Rev. A
2243:
2230:
2205:
2199:
2193:
2168:
2164:
2158:
2147:. Retrieved
2140:the original
2135:
2132:JETP Letters
2131:
2118:
2096:(5–6): 321.
2093:
2089:
2079:
2046:
2042:
2036:
1993:
1989:
1985:
1981:
1975:
1942:
1938:
1925:
1900:
1896:
1890:
1855:
1851:
1845:
1765:applications
1739:
1720:beam into a
1715:
1694:
1673:
1656:
1638:birefringent
1634:birefringent
1631:
1627:
1271:
1267:
1028:
1024:
742:
738:
727:
722:
431:
258:
245:
235:
228:
218:
206:polarization
203:
177:
158:
156:
149:
135:
117:
114:Introduction
101:
94:
88:
77:polarization
74:
62:
55:
39:
37:
2250:(5): 4064.
2208:(9): S951.
1996:(1): 52–4.
1641:the Earth.
2944:Categories
2832:Phorbitech
2748:(4): 282.
2701:: 79480L.
2695:Proc. SPIE
2293:(3): 221.
2171:(5): 985.
2149:2011-09-27
1988:spanner".
1986:An optical
1903:(7): 585.
1858:(5): 449.
1837:References
1726:astigmatic
187:collimated
93:(SAM) and
2772:118624858
2755:1009.5128
2646:1204.5781
2415:0905.0562
2353:0712.0099
2303:CiteSeerX
2222:119660334
2006:CiteSeerX
1868:CiteSeerX
1510:ϕ
1507:∂
1503:∂
1496:∗
1450:∑
1446:∫
1440:ω
1423:ε
1416:^
1356:−
1323:∫
1317:ω
1303:ϵ
1296:^
1284:≈
1215:∇
1211:×
1196:∗
1174:∫
1147:∑
1140:ω
1124:ε
1090:×
1085:∗
1070:∫
1064:ω
1048:ε
981:⊥
967:∇
963:×
948:⊥
926:∫
905:∑
895:ε
866:⊥
856:×
851:⊥
836:∫
827:ε
818:⊥
789:⊥
760:⊥
735:invariant
628:∇
624:×
596:∫
569:∑
559:ε
527:×
514:∫
505:ε
407:ϵ
314:×
301:×
293:∫
284:ε
214:wavefront
199:beam axis
126:but also
81:wavefront
2855:Archived
2815:16389814
2723:16918229
2679:22554538
2671:22714347
2617:19484105
2440:52203556
2386:15600569
2378:16712234
2325:19784282
2272:53501305
2071:11863722
2028:18183100
1967:10060128
1773:See also
246:internal
236:external
225:momentum
179:paraxial
170:magnetic
166:electric
145:pressure
128:momentum
63:twisting
56:spinning
2795:Bibcode
2703:Bibcode
2651:Bibcode
2595:Bibcode
2568:9913630
2548:Bibcode
2518:9906912
2498:Bibcode
2463:Bibcode
2420:Bibcode
2358:Bibcode
2295:Bibcode
2252:Bibcode
2173:Bibcode
2098:Bibcode
2051:Bibcode
1998:Bibcode
1947:Bibcode
1905:Bibcode
1897:Physica
1860:Bibcode
1852:Physica
1697:q-plate
1683:Q-plate
717:is the
425:is the
229:orbital
191:Fourier
99:(OAM).
50:of the
2929:
2908:
2889:
2813:
2770:
2721:
2677:
2669:
2615:
2566:
2516:
2438:
2384:
2376:
2323:
2305:
2270:
2220:
2069:
2026:
2008:
1965:
1870:
1558:where
695:where
462:) and
354:where
263:of an
124:energy
61:, or "
44:vector
2955:Light
2768:S2CID
2750:arXiv
2719:S2CID
2675:S2CID
2641:arXiv
2536:(PDF)
2436:S2CID
2410:arXiv
2382:S2CID
2348:arXiv
2268:S2CID
2240:(PDF)
2218:S2CID
2143:(PDF)
2128:(PDF)
1935:(PDF)
1800:Light
731:gauge
108:force
52:light
42:is a
31:beam.
2927:ISBN
2906:ISBN
2887:ISBN
2847:ICFO
2811:PMID
2699:7948
2667:PMID
2613:PMID
2564:PMID
2514:PMID
2374:PMID
2321:PMID
2067:PMID
2024:PMID
1963:PMID
1585:and
774:and
729:not
376:and
168:and
38:The
2803:doi
2760:doi
2711:doi
2659:doi
2603:doi
2556:doi
2506:doi
2471:doi
2459:104
2428:doi
2366:doi
2313:doi
2260:doi
2210:doi
2181:doi
2106:doi
2094:112
2059:doi
2016:doi
1955:doi
1913:doi
1878:doi
1761:or
488:):
464:OAM
438:SAM
70:eye
2946::
2809:.
2801:.
2791:30
2789:.
2766:.
2758:.
2744:.
2717:.
2709:.
2673:.
2665:.
2657:.
2649:.
2637:20
2635:.
2611:.
2601:.
2591:12
2589:.
2585:.
2562:.
2554:.
2544:54
2542:.
2538:.
2512:.
2504:.
2494:45
2492:.
2469:.
2457:.
2434:.
2426:.
2418:.
2406:94
2404:.
2380:.
2372:.
2364:.
2356:.
2344:96
2342:.
2319:.
2311:.
2301:.
2291:17
2289:.
2266:.
2258:.
2248:56
2246:.
2242:.
2216:.
2206:24
2204:.
2179:.
2169:39
2167:.
2136:52
2134:.
2130:.
2104:.
2092:.
2065:.
2057:.
2047:88
2045:.
2022:.
2014:.
2004:.
1994:22
1992:.
1961:.
1953:.
1943:75
1941:.
1937:.
1911:.
1901:10
1899:.
1876:.
1866:.
1854:.
201:.
2935:.
2914:.
2895:.
2817:.
2805::
2797::
2774:.
2762::
2752::
2746:4
2725:.
2713::
2705::
2681:.
2661::
2653::
2643::
2619:.
2605::
2597::
2570:.
2558::
2550::
2520:.
2508::
2500::
2477:.
2473::
2465::
2442:.
2430::
2422::
2412::
2388:.
2368::
2360::
2350::
2327:.
2315::
2297::
2274:.
2262::
2254::
2224:.
2212::
2187:.
2183::
2175::
2152:.
2112:.
2108::
2100::
2073:.
2061::
2053::
2030:.
2018::
2000::
1969:.
1957::
1949::
1919:.
1915::
1907::
1884:.
1880::
1862::
1856:7
1598:R
1594:E
1571:L
1567:E
1546:.
1542:r
1536:3
1532:d
1527:)
1521:i
1517:E
1489:i
1485:E
1478:(
1472:z
1469:,
1466:y
1463:,
1460:x
1457:=
1454:i
1437:i
1434:2
1427:0
1413:z
1404:+
1400:r
1394:3
1390:d
1385:)
1379:2
1374:|
1369:R
1365:E
1361:|
1351:2
1346:|
1341:L
1337:E
1333:|
1327:(
1314:2
1307:0
1293:z
1280:J
1254:.
1250:r
1244:3
1240:d
1235:)
1229:i
1225:E
1220:)
1207:r
1202:(
1189:i
1185:E
1178:(
1169:z
1166:,
1163:y
1160:,
1157:x
1154:=
1151:i
1137:i
1134:2
1128:0
1118:+
1114:r
1108:3
1104:d
1099:)
1094:E
1080:E
1074:(
1061:i
1058:2
1052:0
1042:=
1038:J
1011:.
1007:r
1001:3
997:d
992:)
986:i
977:A
972:)
959:r
954:(
941:i
937:E
930:(
921:y
918:,
915:x
912:=
909:i
899:0
891:+
887:r
881:3
877:d
872:)
861:A
846:E
840:(
831:0
823:=
813:J
784:E
755:A
743:E
739:A
733:-
723:i
704:A
683:,
679:L
675:+
671:S
667:=
663:r
657:3
653:d
648:)
642:i
638:A
633:)
620:r
615:(
609:i
605:E
600:(
591:z
588:,
585:y
582:,
579:x
576:=
573:i
563:0
555:+
551:r
545:3
541:d
536:)
531:A
523:E
518:(
509:0
501:=
497:J
475:L
466:(
449:S
440:(
411:0
385:B
363:E
342:,
338:r
332:3
328:d
323:)
318:B
310:E
305:(
297:r
288:0
280:=
276:J
249:"
239:"
232:"
183:"
162:"
141:"
66:"
59:"
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.