# Optical vortex

Diagram of different modes, four of which are optical vortices. Columns show the helical structures, phase-front and intensity of the beams

An optical vortex (also known as a photonic quantum vortex, screw dislocation or phase singularity) is a zero of an optical field; a point of zero intensity. The term is also used to describe a beam of light that has such a zero in it. The study of these phenomena is known as singular optics.

## Explanation

In an optical vortex, light is twisted like a corkscrew around its axis of travel. Because of the twisting, the light waves at the axis itself cancel each other out. When projected onto a flat surface, an optical vortex looks like a ring of light, with a dark hole in the center. The vortex is given a number, called the topological charge, according to how many twists the light does in one wavelength. The number is always an integer, and can be positive or negative, depending on the direction of the twist. The higher the number of the twist, the faster the light is spinning around the axis.

This spinning carries orbital angular momentum with the wave train, and will induce torque on an electric dipole. Orbital angular momentum is distinct from the more commonly encountered spin angular momentum, which produces circular polarization.[1] Orbital angular momentum of light can be observed in the orbiting motion of trapped particles. Interfering an optical vortex with a plane wave of light reveals the spiral phase as concentric spirals. The number of arms in the spiral equals the topological charge.

Optical vortices are studied by creating them in the lab in various ways. They can be generated directly in a laser,[2][3] or a laser beam can be twisted into vortex using any of several methods, such as computer-generated holograms, spiral-phase delay structures, or birefringent vortices in materials.

## Properties

A Laguerre-Gaussian beam is an optical vortex with a line singularity along the beam axis

An optical singularity is a zero of an optical field. The phase in the field circulates around these points of zero intensity (giving rise to the name vortex). Vortices are points in 2D fields and lines in 3D fields (as they have codimension two). Integrating the phase of the field around a path enclosing a vortex yields an integer multiple of 2π. This integer is known as the topological charge, or strength, of the vortex.

A hypergeometric-Gaussian mode (HyGG) has an optical vortex in its center. The beam, which has the form

${\displaystyle \psi \propto e^{im\phi }e^{-r^{2}},\!}$

is a solution to the paraxial wave equation (see paraxial approximation, and the Fourier optics article for the actual equation) consisting of the Bessel function. Photons in a hypergeometric-Gaussian beam have an orbital angular momentum of . The integer m also gives the strength of the vortex at the beam's centre. Spin angular momentum of circularly polarized light can be converted into orbital angular momentum.[4]

## Creation

Several methods exist to create hypergeometric-Gaussian modes, including with a spiral phase plate, computer-generated holograms, mode conversion, a q-plate, or a spatial light modulator.

• Static spiral phase plate(s) or mirror(s) are spiral-shaped pieces of crystal or plastic that are engineered specifically to the desired topological charge and incident wavelength. They are efficient, yet expensive. Adjustable spiral phase plates can be made by moving a wedge between two sides of a cracked piece of plastic. Off-axis spiral phase mirrors can be used to mode convert high-power and ultra-short lasers.
• Computer-generated holograms (CGHs) are the calculated interferogram between a plane wave and a Laguerre-Gaussian beam which is transferred to film. The CGH resembles a common Ronchi linear diffraction grating, save a "fork" dislocation. An incident laser beam creates a diffraction pattern with vortices whose topological charge increases with diffraction order. The zero order is Gaussian, and the vortices have opposite helicity on either side of this undiffracted beam. The number of prongs in the CGH fork is directly related to the topological charge of the first diffraction order vortex. The CGH can be blazed to direct more intensity into the first order. Bleaching transforms it from an intensity grating to a phase grating, which increases efficiency.
Vortices created by CGH
• Mode conversion requires Hermite-Gaussian (HG) modes, which can easily be made inside the laser cavity or externally by less accurate means. A pair of astigmatic lenses introduces a Gouy phase shift which creates an LG beam with azimuthal and radial indices dependent upon the input HG.
• A spatial light modulator is a computer-controlled electronic liquid-crystal device which can create dynamic vortices, arrays of vortices, and other types of beams by creating a hologram of varying refractive indices.[5] This hologram may be a fork pattern, a spiral phase plate, or some similar pattern with non-zero topological charge.
• Deformable mirror made of segments can be used to dynamically (with a rate of up to a few kHz) create vortices, even if illuminated by high power lasers.
• A q-plate is a birefringent liquid crystal plate with an azimuthal distribution of the local optical axis, which has a topological charge q at its center defect. The q-plate with topological charge q can generate a ${\displaystyle \pm 2q}$ charge vortex based on the input beam polarization.
• An s-plate is a similar technology to a q-plate, using a high-intensity UV laser to permanently etch a birefringent pattern into silica glass with an azimuthal variation in the fast axis with topological charge of s. Unlike a q-plate, which may be wavelength tuned by adjusting the bias voltage on the liquid crystal, an s-plate only works for one wavelength of light.
• At radio frequencies it is trivial to produce a (non optical) electromagnetic vortex. Simply arrange a one wavelength or greater diameter ring of antennas such that the phase shift of the broadcast antennas varies an integral multiple of 2π around the ring.
• Nanophotonic metasurfaces can enable transverse phase modulation to create optical vortices.[6][7] The vortex beams can be generated in either free space[8][9] or on an integrated photonic chip.[10][11][12]

## Detection

An optical vortex, being fundamentally a phase structure, cannot be detected from its intensity profile alone. Furthermore, as vortex beams of the same order have roughly identical intensity profiles, they cannot be solely characterized from their intensity distributions. As a result, a wide range of interferometric techniques are employed.

• The simplest of the techniques is to interfere a vortex beam with an inclined plane wave, which results in a fork-like interferogram. By making a count of the number of forks in the pattern and their relative orientations, the vortex order and its corresponding sign can be precisely estimated.[13]
• A vortex beam can be deformed into its characteristic lobe structure while passing through a tilted lens. This happens as a result of a self-interference between different phase points in a vortex. A vortex beam of order l will be split into n = l + 1 lobes, roughly around the depth of focus of a tilted convex lens. Furthermore, the orientation of lobes (right and left diagonal), determine the positive and negative orbital angular momentum orders.[14]
• A vortex beam generates a lobe structure when interfered with a vortex of opposite sign. This technique offers no mechanism to characterize the signs, however. This technique can be employed by placing a Dove prism in one of the paths of a Mach–Zehnder interferometer, pumped with a vortex profile.[13]

## Applications

There are a broad variety of applications of optical vortices in diverse areas of communications and imaging.

• Extrasolar planets have only recently been directly detected, as their parent star is so bright. Progress has been made in creating an optical vortex coronagraph to directly observe planets with too low a contrast ratio to their parent to be observed with other techniques.
• Optical vortices are used in optical tweezers to manipulate micrometer-sized particles such as cells. Such particles can be rotated in orbits around the axis of the beam using OAM. Micro-motors have also been created using optical vortex tweezers.
• Optical vortices can significantly improve communication bandwidth. For instance, twisted radio beams could increase radio spectral efficiency by using the large number of vortical states.[15][16][17] The amount of phase front ‘twisting’ indicates the orbital angular momentum state number, and beams with different orbital angular momentum are orthogonal. Such orbital angular momentum based multiplexing can potentially increase the system capacity and spectral efficiency of millimetre-wave wireless communication.[18]
• Similarly, early experimental results for orbital angular momentum multiplexing in the optical domain have shown results over short distances,[19][20] but longer distance demonstrations are still forthcoming. The main challenge that these demonstrations have faced is that conventional optical fibers change the spin angular momentum of vortices as they propagate, and may change the orbital angular momentum when bent or stressed. So far stable propagation of up to 50 meters has been demonstrated in specialty optical fibers.[21] Free-space transmission of orbital angular momentum modes of light over a distance of 143 km has been demonstrated to be able to support encoding of information with good robustness.[22]
• Current computers use electronics that have two states, zero and one. Quantum computing could use light to encode and store information. Optical vortices theoretically have an infinite number of states in free space, as there is no limit to the topological charge.[citation needed] This could allow for faster data manipulation. The cryptography community is also interested in optical vortices for the promise of higher bandwidth communication discussed above.
• In optical microscopy, optical vortices may be used to achieve spatial resolution beyond normal diffraction limits using a technique called Stimulated Emission Depletion (STED) Microscopy. This technique takes advantage of the low intensity at the singularity in the center of the beam to deplete the fluorophores around a desired area with a high-intensity optical vortex beam without depleting fluorophores in the desired target area.[23]
• Optical vortices can be also directly (resonantly) transferred into polariton fluids of light and matter to study the dynamics of quantum vortices upon linear or nonlinear interaction regimes.[24]
• Optical vortices can be identified in the non-local correlations of entangled photon pairs.[25]

## References

1. ^ Allen, L.; Beijersbergen, M. W.; Spreeuw, R. J. C.; Woerdman, J. P. (1992). "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes". Phys. Rev. A. 45 (11): 8185–8189. Bibcode:1992PhRvA..45.8185A. doi:10.1103/PhysRevA.45.8185. PMID 9906912.
2. ^ White, AG; Smith, CP; Heckenberg, NR; Rubinsztein-Dunlop, H; McDuff, R; Weiss, CO; Tamm, C (1991). "Interferometric measurements of phase singularities in the output of a visible laser". Journal of Modern Optics. 38 (12): 2531–2541. Bibcode:1991JMOp...38.2531W. doi:10.1080/09500349114552651.
3. ^ Naidoo, Darryl; et al. (2016). "Controlled generation of higher-order Poincaré sphere beams from a laser". Nature Photonics. 10 (5): 327–332. arXiv:1505.02256. Bibcode:2016NaPho..10..327N. doi:10.1038/nphoton.2016.37. S2CID 7737430.
4. ^ Marrucci, L.; Manzo, C; Paparo, D (2006). "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media". Physical Review Letters. 96 (16): 163905. arXiv:0712.0099. Bibcode:2006PhRvL..96p3905M. doi:10.1103/PhysRevLett.96.163905. PMID 16712234. S2CID 15600569.
5. ^ Heckenberg, NR; McDuff, R; Smith, CP; White, AG (1992). "Generation of optical phase singularities by computer-generated holograms" (PDF). Optics Letters. 17 (3): 221–223. Bibcode:1992OptL...17..221H. doi:10.1364/OL.17.000221. PMID 19784282.
6. ^ Devlin, Robert C.; Ambrosio, Antonio; Rubin, Noah A.; Mueller, J. P. Balthasar; Capasso, Federico (2017-11-17). "Arbitrary spin-to–orbital angular momentum conversion of light". Science. 358 (6365): 896–901. doi:10.1126/science.aao5392. ISSN 0036-8075.
7. ^ Meng, Yuan; Liu, Zhoutian; Xie, Zhenwei; Wang, Ride; Qi, Tiancheng; Hu, Futai; Kim, Hyunseok; Xiao, Qirong; Fu, Xing; Wu, Qiang; Bae, Sang-Hoon; Gong, Mali; Yuan, Xiaocong (2020-04-01). "Versatile on-chip light coupling and (de)multiplexing from arbitrary polarizations to controlled waveguide modes using an integrated dielectric metasurface". Photonics Research. 8 (4): 564. doi:10.1364/PRJ.384449. ISSN 2327-9125.
8. ^ Ren, Haoran; Briere, Gauthier; Fang, Xinyuan; Ni, Peinan; Sawant, Rajath; Héron, Sébastien; Chenot, Sébastien; Vézian, Stéphane; Damilano, Benjamin; Brändli, Virginie; Maier, Stefan A.; Genevet, Patrice (2019-07-19). "Metasurface orbital angular momentum holography". Nature Communications. 10 (1): 2986. doi:10.1038/s41467-019-11030-1. ISSN 2041-1723.
9. ^ Yu, Nanfang; Genevet, Patrice; Kats, Mikhail A.; Aieta, Francesco; Tetienne, Jean-Philippe; Capasso, Federico; Gaburro, Zeno (2011-10-21). "Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction". Science. 334 (6054): 333–337. doi:10.1126/science.1210713. ISSN 0036-8075.
10. ^ Guo, Xuexue; Ding, Yimin; Chen, Xi; Duan, Yao; Ni, Xingjie (2020-07-17). "Molding free-space light with guided wave–driven metasurfaces". Science Advances. 6 (29): eabb4142. doi:10.1126/sciadv.abb4142. ISSN 2375-2548. PMC 7439608. PMID 32832643.
11. ^ Meng, Yuan; Chen, Yizhen; Lu, Longhui; Ding, Yimin; Cusano, Andrea; Fan, Jonathan A.; Hu, Qiaomu; Wang, Kaiyuan; Xie, Zhenwei; Liu, Zhoutian; Yang, Yuanmu; Liu, Qiang; Gong, Mali; Xiao, Qirong; Sun, Shulin (2021-11-22). "Optical meta-waveguides for integrated photonics and beyond". Light: Science & Applications. 10 (1): 235. doi:10.1038/s41377-021-00655-x. ISSN 2047-7538.
12. ^ Cai, Xinlun; Wang, Jianwei; Strain, Michael J.; Johnson-Morris, Benjamin; Zhu, Jiangbo; Sorel, Marc; O’Brien, Jeremy L.; Thompson, Mark G.; Yu, Siyuan (2012-10-19). "Integrated Compact Optical Vortex Beam Emitters". Science. 338 (6105): 363–366. doi:10.1126/science.1226528. ISSN 0036-8075.
13. ^ a b Gbur, Greg (2015). "Singular Optics". The Optics Encyclopedia. Wiley. pp. 1–23. doi:10.1002/9783527600441.oe1011. ISBN 9783527600441.
14. ^ Vaity, Pravin; Banerji, J.; Singh, R.P. (2013). "Measuring the topological charge of an optical vortex by using a tilted convex lens". Physics Letters A. 377 (15): 1154–1156. Bibcode:2013PhLA..377.1154V. doi:10.1016/j.physleta.2013.02.030. ISSN 0375-9601.
15. ^ Twisted radio beams could untangle the airwaves
16. ^ Utilization of Photon Orbital Angular Momentum in the Low-Frequency Radio Domain
17. ^ Encoding many channels on the same frequency through radio vorticity: first experimental test
18. ^ Yan, Yan (16 September 2014). "High-capacity millimetre-wave communications with orbital angular momentum multiplexing". Nature Communications. 5: 4876. Bibcode:2014NatCo...5.4876Y. doi:10.1038/ncomms5876. PMC 4175588. PMID 25224763.
19. ^ "'Twisted light' carries 2.5 terabits of data per second". BBC News. 2012-06-25. Retrieved 2012-06-25.
20. ^ Bozinovic, Nenad (June 2013). "Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers". Science. 340 (6140): 1545–1548. Bibcode:2013Sci...340.1545B. doi:10.1126/science.1237861. PMID 23812709. S2CID 206548907.
21. ^ Gregg, Patrick (January 2015). "Conservation of orbital angular momentum in air-core optical fibers". Optica. 2 (3): 267–270. arXiv:1412.1397. Bibcode:2015Optic...2..267G. doi:10.1364/optica.2.000267. S2CID 119238835.
22. ^ Krenn, M; et al. (2016). "Twisted Light Transmission over 143 kilometers". PNAS. 113 (48): 13648–13653. arXiv:1606.01811. Bibcode:2016PNAS..11313648K. doi:10.1073/pnas.1612023113. PMC 5137742. PMID 27856744.
23. ^ Yan, Lu (September 2015). "Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy". Optica. 2 (10): 900–903. Bibcode:2015Optic...2..900Y. doi:10.1364/optica.2.000900. S2CID 52238379.
24. ^ Dominici, L; Dagvadorj, G; Fellows, JM; et al. (2015). "Vortex and half-vortex dynamics in a nonlinear spinor quantum fluid". Science Advances. 1 (11): e1500807. arXiv:1403.0487. Bibcode:2015SciA....1E0807D. doi:10.1126/sciadv.1500807. PMC 4672757. PMID 26665174.
25. ^ Gomes, R. M.; Salles, A.; Toscano, F.; Souto Ribeiro, P.H. (16 July 2009). "Observation of a Nonlocal Optical Vortex". Phys. Rev. Lett. 103 (3): 033602. arXiv:0902.1659. Bibcode:2009PhRvL.103c3602G. doi:10.1103/PhysRevLett.103.033602. PMID 19659278.