Abstract
Thirty years ago, Coullet et al. proposed that a special optical field exists in laser cavities bearing some analogy with the superfluid vortex. Since then, optical vortices have been widely studied, inspired by the hydrodynamics sharing similar mathematics. Akin to a fluid vortex with a central flow singularity, an optical vortex beam has a phase singularity with a certain topological charge, giving rise to a hollow intensity distribution. Such a beam with helical phase fronts and orbital angular momentum reveals a subtle connection between macroscopic physical optics and microscopic quantum optics. These amazing properties provide a new understanding of a wide range of optical and physical phenomena, including twisting photons, spin–orbital interactions, Bose–Einstein condensates, etc., while the associated technologies for manipulating optical vortices have become increasingly tunable and flexible. Hitherto, owing to these salient properties and optical manipulation technologies, tunable vortex beams have engendered tremendous advanced applications such as optical tweezers, highorder quantum entanglement, and nonlinear optics. This article reviews the recent progress in tunable vortex technologies along with their advanced applications.
Introduction
Vortices are common phenomena that widely exist in nature, from quantum vortices in liquid nitrogen to ocean circulation and typhoon vortices and even to spiral galaxies in the Milky Way, manifesting themselves not only in macroscopic matter but also in structured electromagnetic and optical fields. This year is the 30th anniversary of the birth of optical vortices (OVs). In 1989, Coullet et al.^{1} found the vortex solutions of the MaxwellBloch equations and created the concept of OVs, inspired by hydrodynamic vortices. Before the proposal of OVs, the analogy between laser physics and fluids/superfluids was already recognized^{2} as early as 1970 by reducing the laser equations to complex Ginzburg–Landau equations (CGLEs), which constitute a class of universal models describing pattern formation in a vast variety of phenomena such as superconductivity, superfluidity, and BoseEinstein condensation^{3}. Later, many hydrodynamic features, such as chaos, multistability, and turbulence, were analogically studied in optical fields^{4,5,6} and observed in laser systems^{7,8,9}. Among the various hydrodynamic effects, the vortex soliton is quite attractive due to its distinctive structure carrying a singularity^{5,6,7}. Analogous to the flow singularity in a fluid vortex, an optical vortex soliton has a phase singularity that appears as an isolated dark spot possessing the topological charges (TCs) of a helical phase^{5,10}. Novel optical vortex solitons were intensively explored based on CGLEs. For instance, stable vortex solitons^{11} and dissipative vortex solitons with trapping potentials^{12} can be solved by twodimensional CGLEs. Topologically multicharged rotating vortex solitons^{13} and vortex excitation with feedback^{14} in lasers were also studied by CGLEs. Moreover, complicated threedimensional toroidal dissipative vortex solitons^{15} can also be characterized by CGLEs with highorder nonlinearity. In 1992, Allen et al.^{16} proposed the orbital angular momentum (OAM) in vortex beams (VBs) where the OVs propagate in paraxial beams, which unveiled a new understanding of the connection between macroscopic optics and quantum effects. As a typical representative of OVs, a VB has become a classical tool to study the properties of OVs because its generation can be easily realized in the laboratory^{17}. VBs characterized by Hilbert factor exp(iℓθ), e.g., the Laguerre–Gaussian (LG) modes, can carry OAM equivalent to ℓℏ per photon (ℓ is an integer number), and this angular momentum (AM) can be much greater than the spin angular momentum (SAM) related to the photon spin^{10}. The general results of these investigations created a new chapter of modern optics, i.e., singular optics^{18}, which is a great leap forward in the development of traditional optics.
In the first 10 years, 1989–1999, the studies on OVs mainly focused on establishing fundamental theories and exploring basic physical phenomena, paving the way for further studies of the light–matter interaction, topological structures, and quantum nature of light. For instance, the dynamics of transverse pattern formation^{5,6}, the interaction and OAM transfer between OVs and particles^{19,20,21}, vortex solitons in a nonlinear medium^{22,23,24}, nonlinear OAMfrequency transformation^{25,26}, the topological phase in OVs^{27}, the rotational Doppler effect^{28}, and multisingularity arrays or vortex crystals^{5,29} were thoroughly studied. These novel theories lay the foundation for extending further widespread applications by using the unique properties of OVs.
In the second 10 years, 1999–2009, with the development of OAM manipulation, tremendous new applications rapidly emerged. In 2001, Zeilinger’s group^{30} realized the OAMentangled photon pair, bringing OVs or twisted photons into quantum applications^{31}. In 2002, Dholakia’s group trapped particles with controlled rotation^{32} and a threedimensional structure^{33} by VBs, expanding the applications of optical tweezers^{34}. In 2003, Harwit^{35} demonstrated astrophysical OAM light generation and its applications in astrometry. In 2004, Zhuang^{36} used VBs as tweezers to assemble DNA biomolecules, opening up biomedical applications of OVs. In 2005, RitschMarte’s group^{37} used OAM in microscopy and imaging, and Tamburini et al.^{38} reported a superdiffractionlimit imaging approach using OAM. In 2008, Barreiro et al.^{39} presented a coding technology using OAM, giving VBs great advantage for use in optical communications. During the decade, OVs were extended to almost every field of advanced optics.
In the last 10 years, 2009–2019, vortex and OAM applications have made many breakthroughs in rapid succession. In 2010, the optical lattice in farfield diffraction of OVs was unveiled as a very prompt and handy way to detect the TC^{40}. In 2011, Capasso’s group^{41} proposed the generalized laws of reflection and refraction, guiding OV generation in nanoscale metasurfaces. In 2012, OAM beams were directly generated in a nanoscale emitter^{42}. In 2013, Willner’s group demonstrated terabitscale highcapacity optical communication via OAM multiplexing in both free space^{43} and fibres^{44}. In 2016, Zeilinger’s group^{45} generated extreme OAM states of over 10,000ℏ and realized quantum entanglement of these states. In the recent three years, increasing numbers of tunable properties of OVs have been flexibly controlled at the nanoscale, including SAM–OAM conversion for classical^{46} and quantum light^{47}, tunable wavelength from visible^{48,49} to Xray light^{50}, ultrabroadband tunable OAM^{51}, and tunable chirality^{52}. Moreover, the timevarying OAM was recently revealed in extremeultraviolet VB with timedelaytunable high harmonic generation^{53}. To date, OVs have brought about numerous innovations in various fields and are still enabling great novelties with improved tunability.
Throughout the roadmap summarized above and depicted in Fig. 1, we can divide the 30year development into three stages: the first 10 years, the fundamental theories stage; the second, the application development stage; and the third, the technology breakthrough stage. These three stages share one common theme of pursuing improved tunability of OVs because the realization of a broader tunable range of OVs can always benefit the birth of new applications. Thus, we propose the tunability of OVs as a better way to describe stateoftheart achievements of OVs. Traditionally, tunable light always means that the wavelength can be tuned and sometimes means that the pulse width can be tuned for a pulsed laser; however, the tunability of OVs should be expanded to more dimensions due to their exotic properties. The tunability of OVs includes not only the spectral and temporal tunability but also the OAM, chirality, TC, and singularitydistribution tunability. We present reviews in succession on the historical progress in the new tunable methods of OVs driven by the fundamental theories and then the numerous novel applications engendered by the improved tunability of OVs. In the “Properties of OVs” section, we review the fundamental theories and properties of OVs, providing a better understanding of the corresponding applications enabled by the unique properties. In “Progress in vortex generation, tuning, and manipulation” section, we review the generation and manipulation methods of OVs, developed from tuning the TC of a single singularity to controlling a multisingularity array, including wavelength, temporal, and OAMtuning technologies. In “Advanced applications of tunable VBs” section, we comprehensively review various advanced applications derived from vortex manipulation. Concluding remarks and prospects are given in the “Conclusions and perspectives” section.
Properties of OVs
Singularity and topological charge
The salient properties of OVs are mostly related to the topological phase structure. Early in the 1970s before OVs were first observed, the topological structure in the wave phase was already under study. Nye and Berry^{54} demonstrated that wave trains with dislocations could induce a vortex structure where a singularity could be solved in the wave equation, which laid the foundation for the study of vortices in air, water, and even light waves, pushing the discovery of OVs. To understand the profound topology in a plain way, we can refer to a familiar artwork exhibiting a similar structure. Escher’s painting Ascending and Descending shows an impossible scenario where the stairs are ascending clockwise yet have a seamless connection to their origin after a roundtrip, which is an artistic implementation of the Penrose Stairs^{55}, as illustrated in Fig. 2. This structure is impossible in real space but possible in phase space. If the phase angle continually increases clockwise along a closed loop from 0 to 2πℓ and returns to the origin, where the integer ℓ is called the TC, the angle zero is exactly equal to 2πℓ, forming a continuous phase distribution along the closed loop, similar to the topology of the wellknown Möbius strip^{56}. The centre spot of the closed loop where the phase cannot be defined is a phase singularity. The definition of the TC of a singularity for the phase distribution \(\varphi\) is given by:
where C is a tiny closed loop surrounding the singularity. For the light field with phase distribution exp(iℓθ) carrying OAM of ℓℏ per photon, the TC of the centre phase singularity is ℓ. The effect of TC is actually commonly seen in our daily life, e.g., the time distribution on earth has a singularity at the North Pole with a TC of 24 h, the duration that the earth takes to rotate one cycle. The continuous phase along the closed loop results in an integer TC. However, as a peculiar case, a noninteger TC was also experimentally and theoretically investigated in OVs^{57,58}. A phase singularity with a certain TC is a representation of a very simple vortex soliton yet acts as an important unit element in that more complex hydrodynamic vortices with chaos, attractors, and turbulence can be seen as the combination of a set of various singularities. This basic description is widely applicable to air^{53}, water^{4}, light^{1}, electron^{59}, and neutron^{60} vortex fields.
Orbital angular momentum and vortex beams
A VB is a paraxial light beam possessing Hilbert factor exp(iℓθ) and carrying OVs along the propagation axis. OVs are not restricted to VBs, yet as typical OVs, VBs carrying OAM, also called OAM beams, are almost the most attractive form of OVs due to their unique quantumclassicalconnection properties. There are already many review articles on OAM, especially on vortex generation^{61,62}, OAM on metasurfaces^{63}, and basic OAM theories and applications^{64,65}. However, few studies have focused on vortex tunability, which is the main theme of this article. For the introduction of basic theories of OAM, previous reviews usually used the wellknown Poynting picture to describe the AM of the photon^{66,67}, which leads to some difficulties, such as complex expressions of OAM and SAM, incompatibility with quantum optics, and the Abraham–Minkowski dilemma^{68}. Here, we review the recently proposed new theory of the canonical picture^{69,70}, which can overcome these difficulties, to introduce basic properties of OAM. The canonical momentum of light is represented as
where H is the magnetizing field. Gaussian units with \(g = (8\pi \omega )^{  1}\), \(\tilde \varepsilon = \varepsilon + \omega {\mathrm{d}}\varepsilon {\mathrm{/d}}\omega\), and \(\tilde \mu = \mu + \omega {\mathrm{d}}\mu {\mathrm{/d}}\omega\) are used. The canonical SAM and OAM densities are expressed as
The total AM of light is J = S + L. For a light beam, a rotating polarization leads to SAM, while a rotating wavefront leads to OAM. Consider a VB propagating along the zaxis:
The average SAM and OAM can be derived as^{69,70}
where the power density \(W = \frac{{g\omega }}{2}\left( {\tilde \varepsilon \left {\mathbf{E}} \right^2 + \tilde \mu \left {\mathbf{H}} \right^2} \right)\) and \(\sigma = \frac{2{{\rm Im}({m})}}{{1 + \left m \right^2}}\). σ = +1 (−1) and 0 correspond to left (right) circularly polarized light and linearly polarized light, respectively. Thus, Eq. (4) reveals that left (right) circularly polarized light carries an SAM of + ℏ (−ℏ) per photon; the light with Hilbert factor exp(iℓθ) carries an OAM of ℓℏ (ℓ = 0, ±1, ±2, …) per photon, where “ ± ” reveals the chirality of the vortex, as demonstrated in Fig. 3. This is consistent with the AM quantization in quantum optics, i.e., the eigenvalues of SAM and OAM for the photon eigenstate yield \(\hat L_z\left \psi \right\rangle = \ell \hbar \left \psi \right\rangle\) and \(\hat S_z\left \psi \right\rangle = \sigma \hbar \left \psi \right\rangle\). Therefore, the phase factor exp(iℓφ) provides a basic frame of VBs.
Polarization and vector vortices
The previous part focuses on the scalar light field, where the polarization is separable from the space. In scalar vortices, there are topological spatial phase structures, but the polarization is unchanged; e.g., Fig. 4a shows that a circularly polarized OV can be expressed as the product of a spatially varying vortex phase state and a circular polarization state^{71}. If the polarization state has a spatially varying vector distribution forming vortexlike patterns, then the corresponding optical field is called polarization vortices or vector vortices, and the corresponding singularity is called a polarization singularity or a vector singularity^{72,73}. Based on the various topological disclinations of polarization, vector vortices can be categorized into many types, such as Cpoint, Vpoint, lemons, star, spider, and web, according to the actual vector morphology^{74}. In contrast to the phase vortices carrying OAM, the vector vortices are always related to a complex SAMOAM coupling; e.g., Fig. 4b shows a spiderlike vector OV formed by the superposition of opposite phase variations and opposite circular polarizations, where the total OAM is zero due to the sum of the two opposite phase variations but there is a complex SAM entangled with the space^{71}.
Classical models of OVs
LG and Hermite–Laguerre–Gaussian modes
LG modes with circular symmetry are the earliest reported VBs carrying OAM^{16} and can be included in the general family of Hermite–Laguerre–Gaussian (HLG) modes with elliptical vortices^{75,76,77}, thus accommodating the transform from the HG to LG mode, which has recently played increasingly important roles because the exploration of the more general structure of OVs always leads to novel applications:
where HL_{n,m}(·) is a Hermite–Laguerre (HL) polynomial^{75}, \({r=(x,y)^{\mathrm{T}}} = {(r\,\cos{\varphi}, r\,\sin {\varphi})^{\mathrm{T}}}\), \(R(z) = (z_R^2 + z^2)/z\), \(kw^2(z) = 2(z_R^2 + z^2)/z_R\), \(\vartheta (z) = \arctan (z/z_R)\), and z_{R} is the Rayleigh range. For α = 0 or π∕2, the HLG_{n,m} mode is reduced to the HG_{n,m} or HG_{m,n} mode. For α = π∕4 or 3π∕4, the HLG_{n,m} mode is reduced to LG_{p,±ℓ} mode [\(p = \min \left( {m,n} \right)\), \(\ell = m  n\)]. For the other interposed states, the HLG mode has multiple singularities with a total TC of ℓ. As illustrated in Fig. 5, the LG_{p,ℓ} mode can be decomposed into a set of Hermite–Gaussian (HG) modes^{16,17}:
which also interprets the transformation to an LG_{p,ℓ} mode from an HG_{n,m} mode through an astigmatic mode converter (AMC)^{17}.
HelicalInce–Gaussian and singularities hybrid evolution nature mode
The InceGaussian (IG) mode^{78} is the eigenmode of the paraxial wave equation (PWE) separable in elliptical coordinates (ξ, η)^{79}:
where C^{e,o} are normalization constants (the superscripts e and o refer to even and odd modes), \(I_{u,v}^{e,o}\left( { \cdot ,\epsilon } \right)\) are the even and odd Ince polynomials, with \(0\ <\ v\ <\ u\) for even functions, \(0\ <\ u\ <\ v\) for odd functions, and \(\left( {  1} \right)^{u  v} = 1\) for both, and \(\epsilon\in \left( {\left. {0,\infty } \right)} \right.\) is the eccentricity. The special superposition of these modes can form a multisingularity array with OAM, named the helicalIG (HIG) modes^{80,81,82}:
which carries multiple singularities with unit TC, having a total TC of v. Sharing the singularities hybrid evolution nature (SHEN) of the HIG and HLG modes, the SHEN mode is a very general family of structured Gaussian modes including the HG, LG, HLG, and HIG modes, the expression of which is^{83}
The SHEN mode is reduced to the HIG mode when β = ±π/2, to the HLG mode when γ = 0, to the HG mode when (β, γ) = (0,0) or (π, 0), and to the LG mode when (β, γ) = (±π/2, 0). In addition, there is a graphical representation, the socalled SHEN sphere, to visualize the topological evolution of multisingularity beams. Thus, the SHEN mode has great potential to characterize more general structure beams.
Bessel and Mathieu modes
Using the nondiffraction assumption in solving the PWE, we can also solve a set of eigenmodes. Under separable conditions in circular coordinates, the Bessel mode can be obtained as^{84}
Bessel beams with ℓ ≠ 0 are VBs carrying ℓℏ OAM. Another nondiffraction solution separable in elliptical coordinates is the Mathieu modes^{85},
where C_{m} and S_{m} are normalization constants, Je_{m} and Jo_{m} are radial Mathieu functions, and ce_{m} and se_{m} are angular Mathieu functions. Analogous to deriving the HIG mode, a helical Mathieu (HM) beam^{86} can carry multiple singularities and complex OAM^{87}.
Highorder Bessel and HM beams are often called nondiffractive VBs, whose unique properties have been extended to a great number of applications, such as particle assembly and optical communication^{88,89}.
SU(2) geometric modes
When a resonator cavity fulfils the reentrant condition of a coupled quantum harmonic oscillator in SU(2) Lie algebra^{90}, the laser mode undergoes frequency degeneracy with a photon performing as an SU(2) quantum coherent state coupled with a classical periodic trajectory^{91}, which is called an SU(2) geometric mode (GM)^{92}. The frequency degeneracy means that \({\mathrm{\Delta }}f_{\mathrm{T}}/{\mathrm{\Delta }}f_{\mathrm{L}} = P/Q = {\mathrm{\Omega }}\) should be a simple rational number, where P and Q are two coprime integers, and \({\mathrm{\Delta }}f_{\mathrm{T}}\)(\({\mathrm{\Delta }}f_{\mathrm{L}}\)) is the longitudinal (transverse) mode spacing. The wavepacket function of a planar GM is given by^{92}
where phase ϕ_{0} is related to the classical periodic trajectory. \(\psi _{n,m,s}^{\left( {{\mathrm{HG}}} \right)}\) represents the HG_{n,m} mode considering the frequencydependent wavenumber \(k_{n,m,s} = 2\pi f_{n,m,s}/c\), where \(f_{n,m,s} = s \cdot \Delta f_{\mathrm{L}} + \left( {n + m + 1} \right) \cdot \Delta f_{\mathrm{T}}\). If the HG bases are transformed into LG bases, then the circular GM is obtained^{92}:
where \(\psi _{p,\ell ,s}^{\left( {{\mathrm{LG}}} \right)}\) represents the LG_{p,ℓ} mode considering the frequencydependent wavenumber. The vortex circular GM has many unique properties, such as an exotic 3D structure, multiple singularities, and fractional OAM^{92,93}. Note that there are other types of SU(2) modes related to OAM with special properties, such as Lissajous modes^{94}, trochoidal modes^{95}, polygonal VBs^{96}, and SU(2) diffraction lattices^{97} as shown in Fig. 6d, e.
The above forms are classical VBs in free space, which are just optical modes carrying OAM. In addition, there are OVs that are formed by nonOAM beams, as reviewed in the following.
Optical Möbius strips
A direct idea is to arrange the optical parameter into the form of Möbius strips, one of the classical topological models. This type of OV is called an optical Möbius strip (OMS). A simple vortex phase with integer TC can be seen as a phase OMS. In addition to phase vortices, more OMSs can be obtained by arranging the polarization: the major and minor axes of the polarization ellipses that surround singular lines of circular polarization in threedimensional optical ellipse fields can be organized into an OMS, as theoretically proposed^{98,99} and experimentally observed^{49}. Currently, multitwist OMSs can be controlled in both paraxial and nonparaxial vector beams^{56,100}. By combining other spatial and optical parameters into OMSs, more complex structures, such as 3D solitons and topological knots, can be proposed for OVs^{101}.
Vortex knots
The vortex core of an OV can not only be distributed along the propagation axis of a beam but also form closed loops, links and knots embedded in a light field^{102}. As a new form of OVs, vortex knots have stimulated many experimental observation and theoretical studies on the dynamics of knotted vortices^{102,103}. Vortex knots can also show many homologies, such as pigtail braid and Nodal trefoil knots^{104} as shown in Fig. 7c–f. Currently, researchers have realized the isolated manipulation and temporal control of optical vortex knots^{104,105}.
There are many other forms of OVs that cannot be fully covered in this paper. For instance, there are many freespace VB modes that carry OVs and OAM, such as elegant HLG beams^{106}, Airy beams^{107}, Pearcey beams^{108}, and parabolic beams^{109}. There are many morphologies of the nonbeam spatial distribution of OVs with singularities fractality^{110}. It is highly expected that many new formations of OVs will be reported and investigated in future explorations.
Properties of VBs
Reflection and refraction
The reflection of a VB generally does not satisfy the classical reflection law, i.e., the angle of incidence θ_{i} does not equal the angle of reflection θ_{r}. Instead, the reflected light has a spatial deflection effect related to the OAM of the VB^{111}. The difference between θ_{i} and θ_{r} is related to the OAM of the beam, obeying the generalized law of reflection^{41}
where λ and ϕ are the wavelength and phase of the light beam, respectively, and n is the refractive index of the medium. In addition, the refraction of VBs does not satisfy Snell’s law, i.e., n_{t}sinθ_{t} ≠ n_{i}sinθ_{i}. The refraction is related not only to the angles of incidence and refraction (θ_{i} and θ_{t}) and the refractive indices but also to the OAM, obeying the generalized law of refraction^{41}
Interference
For conventional laser beams, the equalinclination interference pattern is equispaced fringes, and the equalthickness interference pattern is Newton’s rings. However, for a VB, the pattern of equalinclination interference with a plane wave is not equispaced fringes but fringes with bifurcation at the singularity of the vortex, and the morphology of the bifurcation is related to the OAM of the beam^{66}. The equalthickness interference pattern of a VB with a plane wave is not Newton’s rings but spiral stripes extending outward from the vortex singularity, the number of which is related to the OAM^{112}. The selfinterference pattern can also show some bifurcation fringes^{112}. These special interference fringes can be used in detection and measurement methods of vortices.
Diffraction
VBs have unique diffraction properties, the aperture diffraction patterns of which are coupled with the actual OAM. Since Hickmann et al.^{40} unveiled in 2010 the exotic lattice pattern in triangularaperture farfield diffraction of VBs, it has been used as an effective method for OAM detection and measurement of femtosecond vortices^{113}, noninteger charge vortices^{114}, and elliptical VBs^{115}. Many other unique farfield diffraction patterns were investigated through a slit^{116}, a square aperture^{117}, a diamondshaped aperture^{118}, a circular aperture^{119}, an offaxis circular aperture^{120}, an isosceles right triangular aperture^{121}, a sectorial screen^{122}, and so on. The Fresnel diffraction of VBs was also studied^{123}. Some special VBs, such as vector VBs^{124} and SU(2) VBs^{97}, can even bring about special lattice structures in diffraction patterns. These special diffraction patterns can be used in vortex detection and measurement methods.
Polarization
The polarization states of conventional beams can be represented on the Poincaré sphere. VBs can have complex transverse structures involving polarization vortices. Upon combining structured polarization with VBs, the vector VBs can demonstrate more amazing properties and more extensive applications^{74}. To characterize a classical family of vector VBs, Holleczek et al. proposed a classicalquantumconnection model to represent cylindrically polarized beams on the Poincaré sphere^{125}; this model was then extended to the highorder Poincaré sphere (HPS)^{126}, which can reveal SAMOAM conversion and more exotic vector beams, including radial and azimuthal polarization beams. In an experiment, controlled generation of HPS beams was realized^{127} as illustrated in Fig. 8f. As an improved formation of the HPS, the hybridorder Poincaré sphere was theoretically proposed^{128}, and the corresponding experimental controlled generation methods were also presented^{129,130}.
Quantum properties
Twisted photons^{31} are associated with the quantum behaviour of macroscopic VBs. Akin to the conventional Heisenberg uncertainty, there is also the formation of uncertainty for twisted photons; i.e., the product of the uncertainties in the angle and the OAM is bounded by Planck’s constant, ΔϕΔL ≥ ℏ/2^{131,132}. The general Fourier relationship between the angle and the OAM of twisted photons was also investigated^{133}. In contrast to the polarizationentangled state with two dimensions, the OAMentangled state can be high dimensional as \(\left \Psi \right\rangle {\mathrm{ = }}\mathop {\sum}\limits_\ell {c_\ell \left \ell \right\rangle _{\mathrm{A}}\left {  \ell } \right\rangle _{\mathrm{B}}}\)^{134}. Combining the polarization and OAM of the photon, more complex SAM–OAM entangled photon pairs were realized^{47,135}. There are many other new quantum properties related to OAM beams, such as the spin–orbit interaction^{136,137,138}, the Hanbury–Brown–Twiss effect^{139}, quantum interference^{140,141}, and the spin Hall effect^{142,143}.
Measurements of OVs
As mentioned above, OVs can be measured by adopting the interference and diffraction properties of VBs. Counting the stripes and lattices in the special interferogram and diffraction patterns serves as a toolkit to measure the TC, OAM, and singularity distributions of corresponding OVs. In addition, for measuring phase vortices, one can use a spatial light modulator (SLM) to carry out phase transformations, reconstructing the target phase to detect the TC and OAM. Typical realizations include the forked diffraction grating detector^{144}, the OAM sorter^{145}, and spiral transformation^{146}. For polarization vortices, the measurement should also consider the detection of the vector field. By introducing a spacevariant structure into a halfwave plate to modulate the polarization, the TC of the polarization singularity in vector VBs can be measured^{147}. For measuring more properties of vector OVs, Forbes’ group introduced quantum measurement methods to classical light and realized more precise measurement of properties such as the nonseparability, SAM–OAM coupling, and vector factors of vector beams^{148,149}, which is widely applicable to more structured OVs.
Progress in vortex generation, tuning, and manipulation
Brief review of vortex generation
The vortex generation methods can be divided into passive vortex generators (converting the fundamental Gaussian beams into VBs by using dynamic or geometric phase plates, metasurfaces, SLMs, etc.) and active vortex laser generators (such as free space or fibre vortex lasers and nanointegrated OAM generators)^{61,112,150}. There have already been some recent reviews on vortex and OAM beam generation^{61,62,63,112,150}. However, a review focused on vortex generation with tunable and multisingularity properties is rare. Hereinafter, we specifically review active vortex generation with tunable properties, including wavelength, temporal, and OAMtunable beams. In particular, the OAMtunable beams include TCtunable and multisingularitytunable beams.
Wavelength and OAMtunable VBs
OAMtuning of VBs can be realized by gainloss control^{151}, offaxis pumping^{92,152}, or the use of a spiral phase plate (SPP)^{153}, a Qplate^{154,155}, or an SLM^{144}. A wavelengthtunable VB can be achieved by designing special liquid crystal devices^{156}, microcavities^{157}, or onchip gratings^{158} or using nonlinear frequency conversion^{159,160}. However, more methods to simultaneously realize wavelength and OAM tuning for novel applications, such as highcapacity optical communication using wavelength and modedivision multiplexing, are still required.
In 2016, Zhang’s group^{161} presented a wavelength and OAMtunable system by employing a tunable fibre laser with an acoustooptic fibre grating with a wavelengthtunable range of 1540–1560 nm and an OAM of ± 1ℏ, as shown in Fig. 9a, b. In 2017, Lyubopytov et al.^{162} designed a microelectromechanical (MEMS) filter system realizing vortex generation with a wavelengthtunable width of 37.5 nm and an OAM of 0~3ℏ. In the same year, Liu et al.^{163} reported a ringpumped Er:YAG solidstate laser generating an 8.4nm wavelengthtunable width and 0~±2ℏ OAMtunable VB. In 2018, Yao et al.^{164} invented a new optical fibre combiner for combining two polarizationcontrollable fundamental modes into a VB with chiral control, obtaining a 30nm wavelengthtunable width and 0~±1ℏ OAM. Our group^{165} proposed solidstate vortex generation utilizing a dualoffaxis pumped ultrawideband Yb:CALGO laser, reaching a wavelengthtunable width of over 10 nm and an OAM range of 0~±15ℏ, as depicted in Fig. 10a. This system was adapted to generate tunable dualwavelength VBs^{166}. Recently, Wang et al.^{167} improved the output efficiency and reduced the threshold of a similar system by using a Zcavity and a birefringent plate in the cavity design, and a 14.5nm wavelengthtunable width and a 0~±14ℏ OAM range were achieved. Wang’s group^{168} designed and implemented a fibrespace coupling vortex laser system, where a wavelengthtunable range of 1530–1565 nm and an OAM of 0~±10ℏ were achieved.
In addition to the abovementioned wavelength and OAMtunable OVs from lasers, there are vortex generators that are known to be intrinsically broadband, which can also be used to obtain wavelength and OAMtunable OVs. For instance, vortex generation from anisotropic solid crystals, both uniaxial and biaxial, can be related to complex SAM–OAM coupling and gain competition effects, leading to the tunability of vector OVs^{72,73,169}. Similar tunable OVs can be generated in chiral liquid crystals in the regime of circular Bragg reflection^{170,171}. Taking advantage of spacevariant anisotropic liquid crystals that can be electrically controlled, wavelength and OAMtunable OVs can be generated within a wide spectral tunable range^{156,172}. Overall, a wider tunable range controlled by a more convenient OV generation method is still required in the current explorations.
Pulsed VBs
Highpeakpower pulsed VBs with different levels of duration have great potential for use in advanced applications, such as optical machining^{173,174}, nonlinear optics^{25,26,48,49}, strongfield physics^{175}, and optical tweezers^{176}. Hereinafter, we review the recent compact and effective pulsed VB sources.
Nanosecondlevel VBs
The generation of nanosecondlevel VBs has always been combined with Qswitched lasers. An earlier method involved selecting the LG modes using an intracavity aperture in a Qswitched solidstate laser^{177}, with which obtaining high mode purity is difficult. In 2013, Kim et al.^{178} used an etalon and a Brewster plate in an acousticoptic Qswitched laser and generated highquality chiralitycontrolled LG beams with an ~250 μJ pulse energy and an ~33 ns duration. In 2016, Zhao et al.^{179} controlled the pump position in an Er:YAG acousticoptic Qswitched laser, generating a nearly 1mJ and 50ns pulsed VB. In 2017, Chen’s group^{180} designed a nanosecond vortex laser system employing a compact Nd:YVO_{4}/Cr^{4+}:YAG passively Qswitched laser with an external AMC. In 2018, He et al.^{181} presented a Cr,Nd:YAG selfQswitched microchip laser to directly generate lowthreshold nanosecond highrepetitionrate vortex pulses without an AMC, where the chirality was controlled by a tilted output mirror. Our group^{182} recently reported a pulsed vortex output directly from a roomtemperature diodepumped Er,Yb:glass microchip laser with an extremely compact structure.
Picosecondlevel VBs
Picosecondlevel VBs have always been realized in a modelocking laser using a narrowband gain medium. In 2011, Koyama et al.^{183} realized a VB in a stressed Ybdoped fibre amplifier seeded by a picosecond modelocked Nd:YVO_{4} laser with a pulse width of 8.2 ps and a peak power of 34.2 kW. However, the master oscillator poweramplifier structure limited the compactness. The discovery of a selfmodelocking effect in neodymiumdoped crystals provided an alternative way to generate picosecond pulses with a quite compact structure^{184}. In 2009, Liang et al.^{185} reported an OV with a pulse width of 20–25 ps and a repetition rate of 3.5 GHz using offaxispumped selfmodelocked Nd:GdVO_{4} lasers with an AMC. In 2013, the same group^{186} improved this system via cavity control and realized the selfmodelocked SU(2) vortex GM with pulse widths of 22.2 ps and 21.1 ps for Ω = 1/4 and Ω = 1/3, respectively^{92}. In 2017, Tung et al.^{187} reported the direct generation of picosecond largeOAM SU(2) vortex pulses in selfmodelocking Nd:YVO_{4} lasers without the help of an AMC, which largely enhanced the compactness. In 2018, Huang et al.^{188} reported an 8.5 ps pulsed VB generated from a modelocked fibre laser, where the polarization could be controlled at arbitrary states on the HPS.
Femtosecondlevel VBs
In contrast to picosecond pulses, femtosecond pulse generation always requires more extreme operating conditions, such as a tightly focused pumping spot, a wide emission band, and a high nonlinear coefficient of the gain medium. Utilizing the external modulation method, flexible temporal shaping of femtosecond VBs was recently realized^{189}. Considering the improvements in compactness and cost, the selfmodelocking laser oscillator scheme is still desirable. In 2013, Chen’s group^{190} reported a selfmodelocked monolithic Yb:KGW laser with a duration of 850 fs and a repetition rate of 22.4 GHz. In 2016, they^{191} improved the system to directly generate a subpicosecond VB carrying OAM by selective pumping. In 2018, Zhang et al.^{192} proposed an allfibre modelocked femtosecond LG_{0,±1} vortex laser with a pulse width of 398 fs. In the same year, Wang et al.^{193} realized direct emission of an ultrafast LG_{0,±1} VB via a ztype cavity design in an SESAM modelocking Yb:QX laser with a pulse width of 360 fs, as shown in Fig. 11c. These structures have recently been improved by using a Yb:KYW oscillator with a defectspot mirror, obtaining a 298fs VB^{194}. Direct generation of sub100fs VBs may be a future target.
Complex OAM manipulation
In addition to TCtunable VBs, beams with multiple singularities can induce exotic tunable OAM. The multisingularity optical field with a vortex array is also known as a vortex lattice or a vortex crystal^{5,29,73}. Strong requirements of multisingularity beams have been put forward because of the boom of special applications such as multiple particle manipulation^{82,195}, 3D displays^{196}, and optical modulation and communication^{197}.
A singularity splitting phenomenon was found in an AMC when the phase matching condition in the AMC was not satisfied^{198,199}. A large number of matrix optics theories were put forward^{200,201}, deriving the HLG mode to describe the controllable generation of a vortex array in the AMC system^{76,77}. Similar to the HLG mode, the HIG mode is also a multisingularity VB, which can be generated in special cavities with selective pumping^{80} and an SLM^{81}. Recently, our group proposed a method of tuning the periodic orbits of an SU(2) GM in a degenerate cavity and further tuning the multisingularity OAM of SU(2) VBs^{202,203,204}, as shown in Fig. 12b. In addition to HIG, HLG and SU(2) VBs, many other multisingularity VBs with special mathematical formulations were generated with different control methods, such as trochoidal VBs^{95}, transversemodelocking vortex lattices ^{202,203}, and polygonal VBs^{96}.
In addition to the above multisingularity modes, people are pursuing more freely tailored methods for arbitrary singularity distributions. SLM modulation combined with a laser source for ondemand modes has been favoured^{205}. Recently, increasing numbers of tailored singularity distributions have been designed and realized via SLMs, such as rectangular and circular multisingularity arrays^{206,207} and arbitrary curvilinear arrays^{208}, and quadrantseparable singularity control^{209}, as presented in Fig. 13.
There are still many novel methods of tuning the multisingularity OAM in more types of exotic OVs. For instance, various OV arrays can be generated by coherent combining technology with digital control^{210}. Infinite scalar and vector OV arrays can be realized in fractional OAM VBs^{211,212}. Ondemand multisingularity VBs can be generated based on the appropriate combination of optical scattering and discrete rotational symmetries of optical isotropic masks^{213} and can be electrically and optically controlled via anisotropic masks^{214,215}.
Despite the numerous multisingularity manipulation methods, the realization of universal and versatile tunability will be the everlasting target in the future.
Advanced applications of tunable VBs
Optical tweezers
Optical tweezers that trap particles using an optical force were proposed by Ashkin^{216}, who won the Nobel Prize in 2018. Benefitting from the study of OAM interactions with matter, OVs were first used in 1995 in optical tweezers and extended to the optical spanner^{19}, where particles can be trapped and driven to move around the singularity. Then, the transformation from optical OAM to mechanical AM was widely studied^{32,33,34}.
With the improvement of vortex tunability, newgeneration tweezers with OVs have shown distinct advantages^{34,217}. As demonstrated in Fig. 14, the novel vortex tweezers can conveniently manipulate not only the spatial positions of particles but also the multiple degrees of freedom of particles, largely extending the automated guiding, assembly, and sorting technology^{217,218}. With the control of multisingularity VBs, many new techniques were designed and applied to trap multiple particles^{82,217,218}, including the fractional optical VB for optical tweezers^{219}. With femtosecond VBs, the tweezers carrying special nonlinear properties can be used to manipulate optical Rayleigh particles^{220}. Furthermore, with femtosecond vector VBs, nonlinearityinduced multiplexed optical trapping and manipulation was designed^{221}, where the number of traps and their orientations could be flexibly controlled. In addition to dielectric particles, metal particles can also be manipulated by novel plasmonic vortex tweezers^{222}, where the vortex field of surface plasmon polaritons can be generated by focusing vector VBs onto a metal film. Plasmonic vortex tweezers as depicted in Fig. 15d, e were shown to be superior in manipulating metal particles with large flexibility^{223}.
Optical communication
In addition to the polarization, amplitude, pulse shape, and wavelength of light, the OAM can be used as an alternative degree of freedom for multiplexing modulation, enlarging the capacity of optical communication^{39}, which is also referred to as mode/spatialdivision multiplexing (MDM/SDM)^{224}. Optical communication by OAM multiplexing has enabled breaking the Tbit level^{43,44}, much beyond the conventional scheme, thus greatly broadening the application scope^{225,226}. With the study of VB propagation in the atmosphere, freespace communication using vortices was gradually improved^{227,228,229}. Furthermore, a sidelobemodulated OV method was proposed for freespace communication with a significant increase in the data transmission capacity^{230}. With the development of multisingularitytunable VBs, the capacity and speed of communication can be further improved^{231}. A variety of special fibres for OAM mode transmission were designed to enable fibrebased vortex communication technology^{232,233}. Recently, a new OAM multiplexing technology using Dammann vortex gratings in fibrefreespace coupled systems realized massive OAM state parallel detection^{234}, offering an opportunity to raise the communication capacity to the Pbit level. OAMmultiplexingbased communication was also demonstrated under many extreme circumstances, such as underwater communication^{235} illustrated in Fig. 16d, highdimensional quantum communication^{236}, and longdistance fibre communication^{237}.
Quantum entanglement
With the recent mature quantum descriptions of twisted photons^{31}, OAM entanglement has engendered plenty of applications^{134}. For instance, highdimensional quantum key distribution (QKD) protocols can be designed based on mutually unbiased bases related to OAM photons^{238}, which motivated highdimensional quantum cryptography for highsecurity communication^{239}. The quantum memory technology for OAM photonic qubits was recently proposed to provide an essential capability for future networks^{240}. Because of the inherent infinite dimension of OAM, the OAM of photons has been successfully used to realize quantum storage in various systems, such as atomic ensembles^{241} and rareearthiondoped crystals^{242}, benefiting highcapacity communication. Highdimensional OAM entanglement was also successfully used in highefficiency digital spiral imaging^{243}. Employing the Hong–Ou–Mandel interference of OAM photons, quantum cloning technology for making copies of unknown quantum states was presented^{244}. With the development of vector VB manipulation, SAM and OAM were combined for quantum communication to further scale the capacity and speed^{245}. Quantum teleportation using OAM can largely improve the technical control of scalable and complicated quantum systems^{246}. To date, the entangled photon system with the highest number of qubits (18 qubits with six entangled photons) with OAM as one degree of freedom has been produced^{247}. Very recently, as a remarkable breakthrough, quantum entanglement between the SAM and OAM states was realized in a metamaterial^{47}.
In addition to scalar phase OVs, vector polarization OVs also have fruitful quantum properties. The nonseparable states between the polarization and space share common properties with the entangled state of photons, which is also called the classical entanglement state^{71,248}. The quantum tomography, Bell parameter, concurrence count, and linear entropy can be realized in vector OVs akin to corresponding quantum measurements^{148,149,248}. Taking advantage of the highdimensional properties of the nonseparable states, quantum walks can be implemented by vector OV modes of light, enlarging the scalable range^{249}. Entanglement beating generated in vector VBs can be used to control spin–orbit coupling in free space^{135}. Highdimensional entanglement has also been utilized in coding quantum channels to improve highcapacity optical communication^{250}, as illustrated in Fig. 17d, e.
Nonlinear optics
With the development of highpower and largeenergy VBs^{92,180,251}, the peak power can exceed the threshold of various nonlinear effects, providing conditions to explore novel nonlinear conversion phenomena related to OAM^{48,49,251}. Conventionally, the development of nonlinear optics was based mainly on the scattering that obeys momentum conservation (Rayleigh scattering, Brillouin scattering, Raman scattering, etc.), and the corresponding development of nonlinear frequency transformation effects (frequency doubling, frequency summing, fourwave mixing, etc.) has benefited a myriad of applications. In the new century, new transverse nonlinear transformation effects have been developed based on AM conservation, such as TC variation during the processes of frequency doubling^{25,26}, summing and mixing^{252,253}, tunable OAM highharmonic transform^{48,49}, and OAM strongfield physics^{175}. Recently, these OAM harmonic generations have been widely applied in nanomaterials for the control of nonlinear phases^{254}, the Pancharatnam–Berry phase^{255} and beam shaping^{256}. In addition, there are many novel physical phenomena coupled with nonlinear OAMfrequency conversion, such as the rotational Doppler effect^{257} and rotational nonlinear absorption^{258}.
Nanotechnology
Due to the rapid development of nanofabrication and increasing demands for nanotechnology applications, nanointegrated onchip vortex generators have emerged for emitting VBs at the nanoscale, such as integrated siliconchipbased VB emitters^{259}, vortex verticalcavity surfaceemitting lasers (VCSELs)^{260}, angular gratings^{42}, micronanoOAM laser emitters^{261}, and various metasurface designs^{262}. Taking advantage of nanoscale VBs, many novel phenomena related to OAM in nanophotonic materials have been demonstrated, such as nondispersive vortices^{263} and SAMtoOAM conversion effects^{46,47}. Combined with new nanomaterials, many vortexemitting materials and devices with unique functions have been invented, such as vector vortex onchip generators^{264} and parallel OAM processors^{265}. Combining quantum technology and nanotechnology, a photonic chip capable of purifying the OAM quantum states was recently produced, which possesses great potential to develop onchip quantum calculation^{266}.
Optical machining
Due to the nature of highorder modes, VBs show weaker capability in conventional machining processes, such as laser cutting and laser punching, than the fundamental Gaussian beam. However, in some special applications, vortex light has distinct advantages. When a metal surface is processed by different vector VBs, various intriguing new patterns can be selectively displayed under light illumination^{267,268}. Moreover, the surface can exhibit different patterns when the illuminated light has different incidence angles^{269}. In addition to the angular sensitivity, a polarizationsensitive surface was fabricated based on a similar technique using vortex processing, i.e., different patterns were exhibited when the surface was illuminated by light with different polarizations^{268}. Utilizing nanophotonics technology, nanoscale VBs were used in nanostructure fabrication. For instance, the chiral nanoneedle structure can be easily fabricated by a perpendicular VB through the transfer of the consequential torque from OAM light to the object^{173,269,270}. Similar methods can produce some other nanostructures, such as helical surface reliefs^{271} and monocrystalline silicon needles^{272}. Recently, highpower ultrashort OAMtunable VBs were combined with femtosecond laser direct writing technology to process more special structures, such as multiwaveguide^{266} and micropipe structures^{174}.
Microscopy and imaging
The unique spiral phase of VBs can be used in phasecontrast microscopy, demonstrating highresolution microimaging^{37}. Applying OAM analysis in the imaging method, the novel digital spiral imaging technique was proposed to improve the resolution^{273}. Currently, imaging using OAM has already realized superdiffractionlimit resolution^{38}. In recent years, a growing number of novel microscopy and imaging technologies using VBs have emerged, reaching increasingly higher resolution. For instance, plasmonic structured illumination microscopy using standing surface plasmon waves induced by OVs was proposed, realizing highresolution widefield imaging^{274}. This microscopy was further improved by using perfect VBs (VBs with a controllable ring radius) to enhance the excitation efficiency and reduce the background noise^{275}. With the development of multisingularity beams, a vortex array was used to harness the pointspread function to realize highresolution farfield microscopy^{276}. Specifically, fractional VBs were also used for precise microscopy to reach sub100nm resolution^{277}. With the advanced vector VBs having a special polarization structure, the superresolution imaging reached an even higher resolution^{278}, as shown in Fig. 18c, d. With the quantum properties of VBs, quantum ghost imaging was combined with twisted photons, opening new routes for imaging techniques^{243}. As a remarkable breakthrough of microscopy using OVs, the stimulated emission depletion (STED) microscopy technique proposed by Willig et al.^{279}, in which the vortex phase is modulated in STED beams to realize superresolution, was awarded the 2014 Nobel Prize in Chemistry.
Biomedicine and chemistry
Using OV tweezers, one can manipulate and assemble some proteins and other biomolecules, greatly advancing the development of structural chemistry and biomedical photonics^{34,36}. Note that VBs and some organic molecules all have chirality, and the chirality of the vortex phase can interplay with that of a biomolecule, which has promoted a number of applications in biomedicine and chemistry^{280}. For instance, VBs can be used to assemble DNA^{36} and resolve enantiomers^{281} due to the chirality coupling effect. By applying this method to chiral metamaterials, novel sensing technology was proposed to detect many enantiomers or biomolecules, such as amino acids, sugars, and nucleotides^{282}. Additionally, the functionalities of transporting subcellular organelles and exerting less photodamage on the trapped particle was developed for vortex tweezers, which have been used in sophisticated singlecell nanosurgery^{283}. The advanced microscopy brought about by VBs was also used for observing biological cell structures with high resolution^{279}. Most recently, vortices were directly generated from organic materials^{284}, with further development of organic illumination and chemical detection technologies expected in the future.
Metrology
Based on the lightmatter interaction through which the OAM of light can be coupled with the mechanical momentum, VBs can be used to detect object motion, including spin motion^{285} and lateral motion^{286}. With recent advances in nanophotonics and nanofabrication, the precision of detection has reached the nanoscale, and VBs can be used for labelfree singlemolecule detection in metamaterials^{287}. Recently, the OAM spectrum, acting as a new powerful tool, was used in optical detection, in which the difference between the OAM spectra of incident and outgoing light revealed the topography of the target^{288}, as depicted in Fig. 19h. Similar OAMspectrum methods have been successfully applied to detect complicated turbulence in the atmosphere^{289} and ocean^{290}. Recently, with the study of the interaction between OVs and plasmonic nanoslits^{291}, VBs have been used to detect the nanostructure on metal films, opening the door for onchip compact OAM detection^{292}. There are also several devices and structures for detecting OAM states. For instance, a virtual rotational antenna structure was designed to generate the rotational Doppler effect, and the signal of the Doppler shift could be detected to reveal the OAM of the corresponding OV^{293}. The onchip plasmonic nanoslit structure can produce different scattering effects for OVs with different TCs, serving as a useful tool for the discrimination of OAM^{294}. Moreover, some ondemand metasurface^{262} and liquid crystal^{170,171,265} devices have shown great potential for detecting OAM, enabling the further development of precise metrology technologies.
Astronomy
OVs not only have been artificially created in laser beams but also naturally exist in the cosmic microwave background^{35}. In 2003, Harwit described astrophysical processes of OAM light generation, including photon scattering and vortex generation in the environments surrounding energetic sources, e.g., masers, pulsars, and quasars^{35}. To make an astronomical survey that took advantage of OVs, an OV coronagraph was designed^{295} and experimentally verified^{296} by Swartzlander’s group, which has made many breakthroughs in astronomical demonstration^{297}. In addition to the scalar vortex masks used in these coronagraph devices, vectorial masks were also implemented in coronagraphs at nearly the same time as Swartzlander’s work in 2005^{298}. With the development of vector OVs, the vortex coronagraph implemented in international groundbased telescope facilities has been based on vectorial vortex masks to obtain higher sensitivity and lower aberrations^{299}. With the recent development of multisingularity tunability, adaptive multiplevortex coronagraph masks have been developed for multiplestar detections^{300,301}. In 2011, Tamburini et al.^{302} reported the OAM light effect around rotating black holes, which provided a new method to detect black holes, as shown in Fig. 20d. Interestingly, astronomical applications are always accompanied by scifi themes, and vortex light has been declared to be a fast, furious and perfect tool for talking to aliens and detecting alien civilizations due to its unique properties^{303}.
Other advances
OVs indeed demonstrate various characteristics, not only as VBs analysed under the paraxial approximation but also as a general spatial singular field with fractality of singularities. In addition, OVs are not restricted to linear space but have been extensively studied in nonlinear media in connection with optical solitons^{7,22,23,24}. Moreover, topological vortex waves can be studied in other spectra in addition to the light field, such as microwave vortices^{304}, acoustic vortices^{305} and Xray vortices^{50}. Vortex electron beams^{59} and neutron beams^{60} with unique OAM properties were also produced and investigated. Very recently, gravitational waves with AM were observed and could be used for trapping and guiding cosmic bodies^{306}. Overall, there are currently numerous promising and amazing applications related to OVs with unlimited possibilities that require further exploration.
Conclusions and perspectives
This review article is dedicated to commemorating the 30th anniversary of the birth of OVs, covering the development history from fundamental theories to tunable vortex techniques and then to widespread scientific applications. We first reviewed the theoretical foundation of OVs and emphasized the unique properties related to OAM, TC, and singularities. Then, we reviewed the recent advances in tunable VBs, where the tunability includes not only wavelength tunability and temporal tunability but also OAM tunability. Recent vortex generation methods with different kinds of tunability were reviewed, revealing the development of optical field manipulation. Taking advantage of the advanced vortex manipulation techniques, widespread novel applications have boomed in the new century. We reviewed the various applications in different branches of science as comprehensively as possible. The development tendency of OVs is a typical example that theories guide new applications and that application demands inspire new theories. To date, OVs are still hot topics and have high potential for both theories and applications.
References
 1.
Coullet, P., Gil, L. & Rocca, F. Optical vortices. Opt. Commun. 73, 403–408 (1989).
 2.
Graham, R. & Haken, H. Laserlight—first example of a secondorder phase transition far away from thermal equilibrium. Z. Phys. 237, 31–46 (1970).
 3.
Aranson, I. S. & Kramer, L. The world of the complex GinzburgLandau equation. Rev. Mod. Phys. 74, 99–143 (2002).
 4.
Coullet, P., Gil, L. & Lega, J. Defectmediated turbulence. Phys. Rev. Lett. 62, 1619–1622 (1989).
 5.
Brambilla, M. et al. Transverse laser patterns. I. Phase singularity crystals. Phys. Rev. A 43, 5090–5113 (1991).
 6.
Brambilla, M. et al. Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics. Phys. Rev. A 43, 5114–5120 (1991).
 7.
Rosanov, N. N., Fedorov, S. V. & Shatsev, A. N. Curvilinear motion of multivortex lasersoliton complexes with strong and weak coupling. Phys. Rev. Lett. 95, 053903 (2005).
 8.
Genevet, P. et al. Bistable and addressable localized vortices in semiconductor lasers. Phys. Rev. Lett. 104, 223902 (2010).
 9.
Barland, S. et al. Observation of “true” optical vortices in a laser system. in Nonlinear Photonics and Novel Optical Phenomena (eds Chen, Z. G. & Morandotti, R.) 195–205 (Springer, New York, NY, 2012).
 10.
Bazhenov, V. Y., Soskin, M. S. & Vasnetsov, M. V. Screw dislocations in light wavefronts. J. Mod. Opt. 39, 985–990 (1992).
 11.
Crasovan, L. C., Malomed, B. A. & Mihalache, D. Stable vortex solitons in the twodimensional Ginzburg–Landau equation. Phys. Rev. E 63, 016605 (2000).
 12.
Mihalache, D. et al. Stable topological modes in twodimensional GinzburgLandau models with trapping potentials. Phys. Rev. A 82, 023813 (2010).
 13.
Fedorov, S. V. et al. Topologically multicharged and multihumped rotating solitons in wideaperture lasers with a saturable absorber. IEEE J. Quantum Electron. 39, 197–205 (2003).
 14.
Paulau, P. V. et al. Vortex solitons in lasers with feedback. Opt. Express 18, 8859–8866 (2010).
 15.
Mihalache, D. et al. Stable vortex tori in the threedimensional cubicquintic GinzburgLandau equation. Phys. Rev. Lett. 97, 073904 (2006).
 16.
Allen, L. et al. Orbital angular momentum of light and the transformation of LaguerreGaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992).
 17.
Beijersbergen, M. W. et al. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt. Commun. 96, 123–132 (1993).
 18.
Dennis, M. R., O’Holleran, K. & Padgett, M. J. Singular optics: optical vortices and polarization singularities. Prog. Opt. 53, 293–363 (2009).
 19.
He, H. et al. Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity. Phys. Rev. Lett. 75, 826–829 (1995).
 20.
Gahagan, K. T. & Swartzlander, G. A. Optical vortex trapping of particles. Opt. Lett. 21, 827–829 (1996).
 21.
Simpson, N. B., Allen, L. & Padgett, M. J. Optical tweezers and optical spanners with Laguerre–Gaussian modes. J. Mod. Opt. 43, 2485–2491 (1996).
 22.
Swartzlander, G. A. Jr. & Law, C. T. Optical vortex solitons observed in Kerr nonlinear media. Phys. Rev. Lett. 69, 2503–2506 (1992).
 23.
Tikhonenko, V., Christou, J. & LutherDaves, B. Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable selffocusing medium. J. Opt. Soc. Am. B 12, 2046–2052 (1995).
 24.
Firth, W. J. & Skryabin, D. V. Optical solitons carrying orbital angular momentum. Phys. Rev. Lett. 79, 2450–2453 (1997).
 25.
Dholakia, K. et al. Secondharmonic generation and the orbital angular momentum of light. Phys. Rev. A 54, R3742–R3745 (1996).
 26.
Courtial, J. et al. Secondharmonic generation and the conservation of orbital angular momentum with highorder LaguerreGaussian modes. Phys. Rev. A 56, 4193–4196 (1997).
 27.
Soskin, M. S. et al. Topological charge and angular momentum of light beams carrying optical vortices. Phys. Rev. A 56, 4064–4075 (1997).
 28.
Courtial, J. et al. Rotational frequency shift of a light beam. Phys. Rev. Lett. 81, 4828–4830 (1998).
 29.
Scheuer, J. & Orenstein, M. Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities. Science 285, 230–233 (1999).
 30.
Mair, A. et al. Entanglement of the orbital angular momentum states of photons. Nature 412, 313–316 (2001).
 31.
MolinaTerriza, G., Torres, J. P. & Torner, L. Twisted photons. Nat. Phys. 3, 305–310 (2007).
 32.
Paterson, L. et al. Controlled rotation of optically trapped microscopic particles. Science 292, 912–914 (2001).
 33.
MacDonald, M. P. et al. Creation and manipulation of threedimensional optically trapped structures. Science 296, 1101–1103 (2002).
 34.
Grier, D. G. A revolution in optical manipulation. Nature 424, 810–816 (2003).
 35.
Harwit, M. Photon orbital angular momentum in astrophysics. Astrophys. J. 597, 1266–1270 (2003).
 36.
Zhuang, X. W. Unraveling DNA condensation with optical tweezers. Science 305, 188–190 (2004).
 37.
Fürhapter, S. et al. Spiral phase contrast imaging in microscopy. Opt. Express 13, 689–694 (2005).
 38.
Tamburini, F. et al. Overcoming the Rayleigh criterion limit with optical vortices. Phys. Rev. Lett. 97, 163903 (2006).
 39.
Barreiro, J. T., Wei, T. C. & Kwiat, P. G. Beating the channel capacity limit for linear photonic superdense coding. Nat. Phys. 4, 282–286 (2008).
 40.
Hickmann, J. M. et al. Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum. Phys. Rev. Lett. 105, 053904 (2010).
 41.
Yu, N. F. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333–337 (2011).
 42.
Cai, X. L. et al. Integrated compact optical vortex beam emitters. Science 338, 363–366 (2012).
 43.
Wang, J. et al. Terabit freespace data transmission employing orbital angular momentum multiplexing. Nat, Photonics 6, 488–496 (2012).
 44.
Bozinovic, N. et al. Terabitscale orbital angular momentum mode division multiplexing in fibers. Science 340, 1545–1548 (2013).
 45.
Fickler, R. et al. Quantum entanglement of angular momentum states with quantum numbers up to 10,010. Proc. Natl Acad. Sci. USA 113, 13642–13647 (2016).
 46.
Devlin, R. C. et al. Arbitrary spinto–orbital angular momentum conversion of light. Science 358, 896–901 (2017).
 47.
Stav, T. et al. Quantum entanglement of the spin and orbital angular momentum of photons using metamaterials. Science 361, 1101–1104 (2018).
 48.
Kong, F. Q. et al. Controlling the orbital angular momentum of high harmonic vortices. Nat. Commun. 8, 14970 (2017).
 49.
Gauthier, D. et al. Tunable orbital angular momentum in highharmonic generation. Nat. Commun. 8, 14971 (2017).
 50.
Lee, J. C. T. et al. Laguerre–Gauss and Hermite–Gauss soft Xray states generated using diffractive optics. Nat. Photonics 13, 205–209 (2019).
 51.
Xie, Z. W. et al. Ultrabroadband onchip twisted light emitter for optical communications. Light Sci. Appl. 7, 18001 (2018).
 52.
Zambon, N. C. et al. Optically controlling the emission chirality of microlasers. Nat. Photonics 13, 283–288 (2019).
 53.
Rego, L. et al. Generation of extremeultraviolet beams with timevarying orbital angular momentum. Science 364, eaaw9486 (2019).
 54.
Nye, J. F. & Berry, M. V. Dislocations in wave trains. Proc. R Soc A Math. Phys. Eng. Sci. 336, 165–190 (1974).
 55.
Penrose, L. S. & Penrose, R. Impossible objects: a special type of visual illusion. Br. J. Psychol. 49, 31–33 (1958).
 56.
Bauer, T. et al. Observation of optical polarization Möbius strips. Science 347, 964–966 (2015).
 57.
Leach, J., Yao, E. & Padgett, M. J. Observation of the vortex structure of a noninteger vortex beam. New J. Phys. 6, 71 (2004).
 58.
Berry, M. V. Optical vortices evolving from helicoidal integer and fractional phase steps. J. Opt. A Pure Appl. Opt. 6, 259–268 (2004).
 59.
Verbeeck, J., Tian, H. & Schattschneider, P. Production and application of electron vortex beams. Nature 467, 301–304 (2010).
 60.
Clark, C. W. et al. Controlling neutron orbital angular momentum. Nature 525, 504–506 (2015).
 61.
Wang, X. W. et al. Recent advances on optical vortex generation. Nanophotonics 7, 1533–1556 (2018).
 62.
Zhu, L. & Wang, J. A review of multiple optical vortices generation: methods and applications. Front. Optoelectron. 12, 52–68 (2019).
 63.
Chen, M. L. M., Jiang, L. J. & Sha, W. E. I. Orbital angular momentum generation and detection by geometricphase based metasurfaces. Appl. Sci. 8, 362 (2018).
 64.
Barnett, S. M., Babiker, M. & Padgett, M. J. Optical orbital angular momentum. Philos. Trans. R Soc. A Math. Phys. Eng. Sci. 375, 20150444 (2017).
 65.
Padgett, M. J. Orbital angular momentum 25 years on [Invited]. Opt. Express 25, 11265–11274 (2017).
 66.
Yao, A. M. & Padgett, M. J. Orbital angular momentum: origins, behavior and applications. Adv. Opti. Photonics 3, 161–204 (2011).
 67.
Milonni, P. W. & Boyd, R. W. Momentum of light in a dielectric medium. Adv. Opt. Photonics 2, 519–553 (2010).
 68.
Nelson, D. F. Momentum, pseudomomentum, and wave momentum: toward resolving the MinkowskiAbraham controversy. Phys. Rev. A 44, 3985–3996 (1991).
 69.
Bliokh, K. Y., Bekshaev, A. Y. & Nori, F. Optical momentum, spin, and angular momentum in dispersive media. Phys. Rev. Lett. 119, 073901 (2017).
 70.
Bliokh, K. Y. & Nori, F. Transverse and longitudinal angular momenta of light. Phys. Rep. 592, 1–38 (2015).
 71.
Karimi, E. & Boyd, R. W. Classical entanglement? Science 350, 1172–1173 (2015).
 72.
Chen, Y. F., Lu, T. H. & Huang, K. F. Observation of spatially coherent polarization vector fields and visualization of vector singularities. Phys. Rev. Lett. 96, 033901 (2006).
 73.
Chen, Y. F. et al. Observation of vector vortex lattices in polarization states of an isotropic microcavity laser. Phys. Rev. Lett. 90, 053904 (2003).
 74.
RosalesGuzmán, C., Ndagano, B. & Forbes, A. A review of complex vector light fields and their applications. J. Opt. 20, 123001 (2018).
 75.
Abramochkin, E. & Alieva, T. Closedform expression for mutual intensity evolution of Hermite–Laguerre–Gaussian Schellmodel beams. Opt. Lett. 42, 4032–4035 (2017).
 76.
Alieva, T. & Bastiaans, M. J. Mode mapping in paraxial lossless optics. Opt. Lett. 30, 1461–1463 (2005).
 77.
Abramochkin, E. G. & Volostnikov, V. G. Generalized HermiteLaguerreGauss beams. Phys. Wave Phenom. 18, 14–22 (2010).
 78.
Bandres, M. A. & GutiérrezVega, J. C. Ince–Gaussian beams. Opt. Lett. 29, 144–146 (2004).
 79.
Bandres, M. A. & GutiérrezVega, J. C. Elliptical beams. Opt. Express 16, 21087–21092 (2008).
 80.
Bandres, M. A. & GutiérrezVega, J. C. Ince–Gaussian modes of the paraxial wave equation and stable resonators. J. Opt. Soc. Am. A 21, 873–880 (2004).
 81.
Bentley, J. B. et al. Generation of helical InceGaussian beams with a liquidcrystal display. Opt. Lett. 31, 649–651 (2006).
 82.
Woerdemann, M., Alpmann, C. & Denz, C. Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams. Appl. Phys. Lett. 98, 111101 (2011).
 83.
Shen, Y. J. et al. Hybrid topological evolution of multisingularity vortex beams: generalized nature for helicalInceGaussian and HermiteLaguerreGaussian modes. J. Opt. Soc. Am. A 36, 578–587 (2019).
 84.
VolkeSepulveda, K. et al. Orbital angular momentum of a highorder Bessel light beam. J. Opt. B Quantum Semiclassical Opt. 4, S82–S89 (2002).
 85.
GutiérrezVega, J. C., IturbeCastillo, M. D. & ChávezCerda, S. Alternative formulation for invariant optical fields: Mathieu beams. Opt. Lett. 25, 1493–1495 (2000).
 86.
LóxpezMariscal, C. et al. Orbital angular momentum transfer in helical Mathieu beams. Opt. Express 14, 4182–4187 (2006).
 87.
ChávezCerda, S. et al. Holographic generation and orbital angular momentum of highorder Mathieu beams. J. Opt. B Quantum Semiclassical Opt. 4, S52–S57 (2002).
 88.
Alpmann, C. et al. Mathieu beams as versatile light moulds for 3D micro particle assemblies. Opt. Express 18, 26084–26091 (2010).
 89.
Zhu, L. & Wang, J. Demonstration of obstructionfree datacarrying Nfold Bessel modes multicasting from a single Gaussian mode. Opt. Lett. 40, 5463–5466 (2015).
 90.
Bužek, V. & Quang, T. Generalized coherent state for bosonic realization of SU(2)Lie algebra. J. Opt. Soc. Am. B 6, 2447–2449 (1989).
 91.
Lin, Y. C. et al. Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes. Phys. Rev. E 85, 046217 (2012).
 92.
Tuan, P. H. et al. Realizing highpulseenergy largeangularmomentum beams by astigmatic transformation of geometric modes in an Nd:YAG/Cr^{4+}:YAG laser. IEEE J. Sel. Top. Quantum Electron. 24, 1600809 (2018).
 93.
Tung, J. C. et al. Exploring vortex structures in orbitalangularmomentum beams generated from planar geometric modes with a mode converter. Opt. Express 24, 22796–22805 (2016).
 94.
Chen, Y. F. et al. Devil’s staircase in threedimensional coherent waves localized on Lissajous parametric surfaces. Phys. Rev. Lett. 96, 213902 (2006).
 95.
Lu, T. H. et al. Threedimensional coherent optical waves localized on trochoidal parametric surfaces. Phys. Rev. Lett. 101, 233901 (2008).
 96.
Shen, Y. J. et al. Polygonal vortex beams. IEEE Photonics J. 10, 1503016 (2018).
 97.
Shen, Y. J., Fu, X. & Gong, M. L. Truncated triangular diffraction lattices and orbitalangularmomentum detection of vortex SU(2) geometric modes. Opt. Express 26, 25545–25557 (2018).
 98.
Freund, I. Optical Möbius strips in threedimensional ellipse fields: I. Lines of circular polarization. Opt. Commun. 283, 1–15 (2010).
 99.
Freund, I. Optical Möbius strips in three dimensional ellipse fields: II. Lines of linear polarization. Opt. Commun. 283, 16–28 (2010).
 100.
Galvez, E. J. et al. Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams. Sci. Rep. 7, 13653 (2017).
 101.
Veretenov, N. A., Fedorov, S. V. & Rosanov, N. N. Topological vortex and knotted dissipative optical 3D solitons generated by 2D vortex solitons. Phys. Rev. Lett. 119, 263901 (2017).
 102.
Leach, J. et al. Vortex knots in light. New J. Phys. 7, 55 (2005).
 103.
Kleckner, D. & Irvine, W. T. M. Creation and dynamics of knotted vortices. Nat. Phys. 9, 253–258 (2013).
 104.
Dennis, M. R. et al. Isolated optical vortex knots. Nat. Phys. 6, 118–121 (2010).
 105.
TemponeWiltshire, S. J., Johnstone, S. P. & Helmerson, K. Optical vortex knots–one photon at a time. Sci. Rep. 6, 24463 (2016).
 106.
Cunzhi, S., Pu, J. X. & ChávezCerda, S. Elegant Cartesian Laguerre–Hermite–Gaussian laser cavity modes. Opt. Lett. 40, 1105–1108 (2015).
 107.
Ellenbogen, T. et al. Nonlinear generation and manipulation of Airy beams. Nat. Photonics 3, 395–398 (2009).
 108.
Ring, J. D. et al. Autofocusing and selfhealing of Pearcey beams. Opt. Express 20, 18955–18966 (2012).
 109.
Bandres, M. A., GutiérrezVega, J. C. & ChávezCerda, S. Parabolic nondiffracting optical wave fields. Opt. Lett. 29, 44–46 (2004).
 110.
O’Holleran, K. et al. Fractality of light’s darkness. Phys. Rev. Lett. 100, 053902 (2008).
 111.
Zhang, L. G. et al. Deflection of a reflected intense vortex laser beam. Phys. Rev. Lett. 117, 113904 (2016).
 112.
Omatsu, T., Miyamoto, K. & Lee, A. J. Wavelengthversatile optical vortex lasers. J. Opt. 19, 123002 (2017).
 113.
de Araujo, L. E. E. & Anderson, M. E. Measuring vortex charge with a triangular aperture. Opt. Lett. 36, 787–789 (2011).
 114.
Mourka, A. et al. Visualization of the birth of an optical vortex using diffraction from a triangular aperture. Opt. Express 19, 5760–5771 (2011).
 115.
Melo, L. A. et al. Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture. Sci. Rep. 8, 6370 (2018).
 116.
Ghai, D. P., Senthilkumaran, P. & Sirohi, R. S. Singleslit diffraction of an optical beam with phase singularity. Opt. Lasers Eng. 47, 123–126 (2009).
 117.
Mesquita, P. H. F. et al. Engineering a square truncated lattice with light’s orbital angular momentum. Opt. Express 19, 20616–20621 (2011).
 118.
Liu, Y. X. et al. Propagation of an optical vortex beam through a diamondshaped aperture. Opt. Laser Technol. 45, 473–479 (2013).
 119.
Ambuj, A., Vyas, R. & Singh, S. Diffraction of orbital angular momentum carrying optical beams by a circular aperture. Opt. Lett. 39, 5475–5478 (2014).
 120.
Taira, Y. & Zhang, S. K. Split in phase singularities of an optical vortex by offaxis diffraction through a simple circular aperture. Opt. Lett. 42, 1373–1376 (2017).
 121.
Bahl, M. & Senthilkumaran, P. Energy circulations in singular beams diffracted through an isosceles right triangular aperture. Phys. Rev. A 92, 013831 (2015).
 122.
Chen, R. S. et al. Detecting the topological charge of optical vortex beams using a sectorial screen. Appl. Opt. 56, 4868–4872 (2017).
 123.
Zhang, W. H. et al. Experimental demonstration of twisted light’s diffraction theory based on digital spiral imaging. Chin. Opt. Lett. 14, 110501 (2016).
 124.
Ram, B. S. B., Sharma, A. & Senthilkumaran, P. Diffraction of Vpoint singularities through triangular apertures. Opt. Express 25, 10270–10275 (2017).
 125.
Holleczek, A. et al. Classical and quantum properties of cylindrically polarized states of light. Opt. Express 19, 9714–9736 (2011).
 126.
Milione, G. et al. Higherorder Poincaré sphere, Stokes parameters, and the angular momentum of light. Phys. Rev. Lett. 107, 053601 (2011).
 127.
Naidoo, D. et al. Controlled generation of higherorder Poincaré sphere beams from a laser. Nat. Photonics 10, 327–332 (2016).
 128.
Yi, X. N. et al. Hybridorder Poincaré sphere. Phys. Rev. A 91, 023801 (2015).
 129.
Liu, Z. X. et al. Generation of arbitrary vector vortex beams on hybridorder Poincaré sphere. Photonics Res. 5, 15–21 (2017).
 130.
Wang, R. S. et al. Electrically driven generation of arbitrary vector vortex beams on the hybridorder Poincaré sphere. Opt. Lett. 43, 3570–3573 (2018).
 131.
FrankeArnold, S. et al. Uncertainty principle for angular position and angular momentum. New J. Phys. 6, 103 (2004).
 132.
Leach, J. et al. Quantum correlations in optical angle–orbital angular momentum variables. Science 329, 662–665 (2010).
 133.
Jha, A. K. et al. Fourier relationship between the angle and angular momentum of entangled photons. Phys. Rev. A 78, 043810 (2008).
 134.
Erhard, M. et al. Twisted photons: new quantum perspectives in high dimensions. Light Sci. Appl. 7, 17146 (2018).
 135.
Otte, E. et al. Entanglement beating in free space through spin–orbit coupling. Light Sci. Appl. 7, 18009 (2018).
 136.
Bliokh, K. Y. et al. Spin–orbit interactions of light. Nat. Photonics 9, 796–808 (2015).
 137.
Cardano, F. & Marrucci, L. Spin–orbit photonics. Nat. Photonics 9, 776–778 (2015).
 138.
Shao, Z. K. et al. Spinorbit interaction of light induced by transverse spin angular momentum engineering. Nat. Commun. 9, 926 (2018).
 139.
MagañaLoaiza, O. S. et al. Hanbury brown and Twiss interferometry with twisted light. Sci. Adv. 2, e1501143 (2016).
 140.
Mohanty, A. et al. Quantum interference between transverse spatial waveguide modes. Nat. Commun. 8, 14010 (2017).
 141.
Zhang, Y. W. et al. Engineering twophoton highdimensional states through quantum interference. Sci. Adv. 2, e1501165 (2016).
 142.
Yin, X. B. et al. Photonic spin Hall effect at metasurfaces. Science 339, 1405–1407 (2013).
 143.
Liu, Y. C. et al. Photonic spin Hall effect in metasurfaces: a brief review. Nanophotonics 6, 51–70 (2017).
 144.
Forbes, A., Dudley, A. & McLaren, M. Creation and detection of optical modes with spatial light modulators. Adv. Opt. Photonics 8, 200–227 (2016).
 145.
Berkhout, G. C. G. et al. Efficient sorting of orbital angular momentum states of light. Phys. Rev. Lett. 105, 153601 (2010).
 146.
Wen, Y. H. et al. Spiral transformation for highresolution and efficient sorting of optical vortex modes. Phys. Rev. Lett. 120, 193904 (2018).
 147.
Liu, G. G. et al. Measurement of the topological charge and index of vortex vector optical fields with a spacevariant halfwave plate. Opt. Lett. 43, 823–826 (2018).
 148.
Ndagano, B. et al. Beam quality measure for vector beams. Opt. Lett. 41, 3407–3410 (2016).
 149.
McLaren, M., Konrad, T. & Forbes, A. Measuring the nonseparability of vector vortex beams. Phys. Rev. A 92, 023833 (2015).
 150.
Forbes, A. Controlling light’s helicity at the source: orbital angular momentum states from lasers. Philos. Trans. R Soc. A Math. Phys. Eng. Sci. 375, 20150436 (2017).
 151.
Qiao, Z. et al. Generating highcharge optical vortices directly from laser up to 288th order. Laser Photonics Rev. 12, 1800019 (2018).
 152.
Lee, C. Y. et al. Generation of higher order vortex beams from a YVO4/Nd:YVO4 selfRaman laser via offaxis pumping with mode converter. IEEE J. Sel. Top Quantum Electron. 21, 1600305 (2015).
 153.
Sueda, K. et al. LaguerreGaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses. Opt. Express 12, 3548–3553 (2004).
 154.
Cardano, F. et al. Polarization pattern of vector vortex beams generated by qplates with different topological charges. Appl. Opt. 51, C1–C6 (2012).
 155.
Marrucci, L. The qplate and its future. J. Nanophotonics 7, 078598 (2013).
 156.
Brasselet, E. Tunable highresolution macroscopic selfengineered geometric phase optical elements. Phys. Rev. Lett. 121, 033901 (2018).
 157.
Mock, A., Sounas, D. & Alù, A. Tunable orbital angular momentum radiation from angularmomentumbiased microcavities. Phys. Rev. Lett. 121, 103901 (2018).
 158.
Zhou, N. et al. Generating and synthesizing ultrabroadband twisted light using a compact silicon chip. Opt. Lett. 43, 3140–3143 (2018).
 159.
Horikawa, M. T. et al. Handedness control in a tunable midinfrared (6.012.5 μm) vortex laser. J. Opt. Soc. Am. B 32, 2406–2410 (2015).
 160.
Abulikemu, A. et al. Widelytunable vortex output from a singly resonant optical parametric oscillator. Opt. Express 23, 18338–18344 (2015).
 161.
Zhang, W. D. et al. Optical vortex generation with wavelength tunability based on an acousticallyinduced fiber grating. Opt. Express 24, 19278–19285 (2016).
 162.
Lyubopytov, V. S. et al. Simultaneous wavelength and orbital angular momentum demultiplexing using tunable MEMSbased FabryPerot filter. Opt. Express 25, 9634–9646 (2017).
 163.
Liu, Q. Y. et al. Wavelength and OAMtunable vortex laser with a reflective volume Bragg grating. Opt. Express 25, 23312–23319 (2017).
 164.
Yao, S. Z. et al. Tunable orbital angular momentum generation using allfiber fused coupler. IEEE Photonics Technol. Lett. 30, 99–102 (2018).
 165.
Shen, Y. J. et al. Wavelengthtunable Hermite–Gaussian modes and an orbitalangularmomentumtunable vortex beam in a dualoffaxis pumped Yb:CALGO laser. Opt. Lett. 43, 291–294 (2018).
 166.
Shen, Y. J. et al. Dualwavelength vortex beam with high stability in a diodepumped Yb:CaGdAlO_{4} laser. Laser Phys. Lett. 15, 055803 (2018).
 167.
Wang, S. et al. Generation of wavelength and OAMtunable vortex beam at low threshold. Opt. Express 26, 18164–18170 (2018).
 168.
Zhou, N., Liu, J. & Wang, J. Reconfigurable and tunable twisted light laser. Sci. Rep. 8, 11394 (2018).
 169.
Fadeyeva, T. A. et al. Spatially engineered polarization states and optical vortices in uniaxial crystals. Opt. Express 18, 10848–10863 (2010).
 170.
Rafayelyan, M., Tkachenko, G. & Brasselet, E. Reflective spinorbit geometric phase from chiral anisotropic optical media. Phys. Rev. Lett. 116, 253902 (2016).
 171.
Kobashi, J., Yoshida, H. & Ozaki, M. Polychromatic optical vortex generation from patterned cholesteric liquid crystals. Phys. Rev. Lett. 116, 253903 (2016).
 172.
Piccirillo, B. et al. Photon spintoorbital angular momentum conversion via an electrically tunable qplate. Appl. Phys. Lett. 97, 241104 (2010).
 173.
Toyoda, K. et al. Using optical vortex to control the chirality of twisted metal nanostructures. Nano Lett. 12, 3645–3649 (2012).
 174.
Yang, L. et al. Direct laser writing of complex microtubes using femtosecond vortex beams. Appl. Phys. Lett. 110, 221103 (2017).
 175.
Zürch, M. et al. Strongfield physics with singular light beams. Nat. Phys. 8, 743–746 (2012).
 176.
Ran, L. L., Guo, Z. Y. & Qu, S. L. Rotational motions of optically trapped microscopic particles by a vortex femtosecond laser. Chin. Phys. B 21, 104206 (2012).
 177.
Ishaaya, A. A. et al. Efficient selection of highorder LaguerreGaussian modes in a Qswitched Nd:YAG laser. IEEE J. Quantum Electron. 39, 74–82 (2003).
 178.
Kim, D. J., Kim, J. W. & Clarkson, W. A. Qswitched Nd:YAG optical vortex lasers. Opt. Express 21, 29449–29454 (2013).
 179.
Zhao, Y. G. et al. 1 mJ pulsed vortex laser at 1645 nm with welldefined helicity. Opt. Express 24, 15596–15602 (2016).
 180.
Chang, C. C. et al. Generating highpeakpower structured lights in selectively pumped passively Qswitched lasers with astigmatic mode transformations. Laser Phys. 27, 125805 (2017).
 181.
He, H. S. et al. Lowthreshold, nanosecond, highrepetitionrate vortex pulses with controllable helicity generated in Cr, Nd:YAG selfQswitched microchip laser. Laser Phys. 28, 055802 (2018).
 182.
Wang, Y. B. et al. Generation of 1535nm pulsed vortex beam in a diodepumped Er, Yb:glass microchip laser. IEEE Photonics Technol. Lett. 30, 891–894 (2018).
 183.
Koyama, M. et al. Power scaling of a picosecond vortex laser based on a stressed Ybdoped fiber amplifier. Opt. Express 19, 994–999 (2011).
 184.
Liang, H. C. et al. Compact efficient multiGHz Kerrlens modelocked diodepumped Nd:YVO_{4} laser. Opt. Express 16, 21149–21154 (2008).
 185.
Liang, H. C. et al. Picosecond optical vortex converted from multigigahertz selfmodelocked highorder HermiteGaussian Nd:GdVO_{4} lasers. Opt. Letters 34, 3842–3844 (2009).
 186.
Liang, H. C. et al. Total selfmode locking of multipass geometric modes in diodepumped Nd:YVO_{4} lasers. Laser Phys. Lett. 10, 105804 (2013).
 187.
Tung, J. C. et al. Exploring the selfmode locking and vortex structures of nonplanar elliptical modes in selectively endpumped Nd:YVO_{4} lasers: manifestation of large fractional orbital angular momentum. Opt. Express 25, 22769–22779 (2017).
 188.
Huang, K. et al. Controlled generation of ultrafast vector vortex beams from a modelocked fiber laser. Opt. Lett. 43, 3933–3936 (2018).
 189.
Bolze, T. & Nuernberger, P. Temporally shaped Laguerre–Gaussian femtosecond laser beams. Appl. Opt. 57, 3624–3628 (2018).
 190.
Zhuang, W. Z. et al. Highpower highrepetitionrate subpicosecond monolithic Yb:KGW laser with selfmode locking. Opt. Lett. 38, 2596–2599 (2013).
 191.
Chang, M. T. et al. Exploring transverse pattern formation in a dualpolarization selfmodelocked monolithic Yb: KGW laser and generating a 25GHz subpicosecond vortex beam via gain competition. Opt. Express 24, 8754–8762 (2016).
 192.
Zhang, Z. M. et al. Generation of allfiber femtosecond vortex laser based on NPR modelocking and mechanical LPG. Chin. Opt. Lett. 16, 110501 (2018).
 193.
Wang, S. et al. Direct emission of chirality controllable femtosecond LG_{01} vortex beam. Appl. Phys. Lett. 112, 201110 (2018).
 194.
Wang, S. et al. Direct generation of femtosecond vortex beam from a Yb:KYW oscillator featuring a defectspot mirror. OSA Contin. 2, 523–530 (2019).
 195.
Woerdemann, M. et al. Advanced optical trapping by complex beam shaping. Laser Photonics Rev. 7, 839–854 (2013).
 196.
Li, X. F. et al. Automultiscopic displays based on orbital angular momentum of light. J. Opt. 18, 085608 (2016).
 197.
Anguita, J. A., Herreros, J. & Djordjevic, I. B. Coherent multimode OAM superpositions for multidimensional modulation. IEEE Photonics J. 6, 7900811 (2014).
 198.
Padgett, M. et al. An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes. Am. J. Phys. 64, 77–82 (1996).
 199.
Courtial, J. & Padgett, M. J. Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes. Opt. Commun. 159, 13–18 (1999).
 200.
O’Neil, A. T. & Courtial, J. Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variablephase mode converter. Opt. Commun. 181, 35–45 (2000).
 201.
Padgett, M. J. & Allen, L. Orbital angular momentum exchange in cylindricallens mode converters. J. Opt. B Quantum Semiclassical Opt. 4, S17–S19 (2002).
 202.
Shen, Y. J. et al. Observation of spectral modulation coupled with broadband transversemodelocking in an Yb:CALGO frequencydegenerate cavity. Chin. Opt. Lett. 17, 031404 (2019).
 203.
Shen, Y. J. et al. Vortex lattices with transversemodelocking states switching in a largeaperture offaxispumped solidstate laser. J. Opt. Soc. Am. B 35, 2940–2944 (2018).
 204.
Shen, Y. J. et al. Periodictrajectorycontrolled, coherentstatephaseswitched, and wavelengthtunable SU(2) geometric modes in a frequencydegenerate resonator. Appl. Opt. 57, 9543–9549 (2018).
 205.
Ngcobo, S. et al. A digital laser for ondemand laser modes. Nat. Commun. 4, 2289 (2013).
 206.
Porfirev, A. P. & Khonina, S. N. Simple method for efficient reconfigurable optical vortex beam splitting. Opt. Express 25, 18722–18735 (2017).
 207.
Ma, H. X. et al. Generation of circular optical vortex array. Ann. Phys. 529, 1700285 (2017).
 208.
Li, L. et al. Generation of optical vortex array along arbitrary curvilinear arrangement. Opt. Express 26, 9798–9812 (2018).
 209.
Wan, Z. S. et al. Quadrantseparable multisingularity vortices manipulation by coherent superposed mode with spatialenergy mismatch. Opt. Express 26, 34940–34955 (2018).
 210.
Hou, T. Y. et al. Spatiallydistributed orbital angular momentum beam array generation based on greedy algorithms and coherent combining technology. Opt. Express 26, 14945–14958 (2018).
 211.
Gbur, G. Fractional vortex Hilbert’s hotel. Optica 3, 222–225 (2016).
 212.
Wang, Y. Y. D. & Gbur, G. Hilbert’s Hotel in polarization singularities. Opt. Lett. 42, 5154–5157 (2017).
 213.
Ferrando, A. & GarcíaMarch, M. A. Analytical solution for multisingular vortex Gaussian beams: the mathematical theory of scattering modes. J. Opt. 18, 064006 (2016).
 214.
Brasselet, E. Tunable optical vortex arrays from a single nematic topological defect. Phys. Rev. Lett. 108, 087801 (2012).
 215.
Barboza, R. et al. Harnessing optical vortex lattices in nematic liquid crystals. Phys. Rev. Lett. 111, 093902 (2013).
 216.
Ashkin, A. Acceleration and trapping of particles by radiation pressure. Phys. Rev. Lett. 24, 156–159 (1970).
 217.
Padgett, M. & Bowman, R. Tweezers with a twist. Nat. Photonics 5, 343–348 (2011).
 218.
Chapin, S. C., Germain, V. & Dufresne, E. R. Automated trapping, assembly, and sorting with holographic optical tweezers. Opt. Express 14, 13095–13100 (2006).
 219.
Tao, S. H. et al. Fractional optical vortex beam induced rotation of particles. Opt. Express 13, 7726–7731 (2005).
 220.
Gong, L. et al. Optical forces of focused femtosecond laser pulses on nonlinear optical Rayleigh particles. Photonics Res. 6, 138–143 (2018).
 221.
Zhang, Y. Q. et al. Nonlinearityinduced multiplexed optical trapping and manipulation with femtosecond vector beams. Nano Lett. 18, 5538–5543 (2018).
 222.
Shen, Z. et al. Visualizing orbital angular momentum of plasmonic vortices. Opt. Lett. 37, 4627–4629 (2012).
 223.
Zhang, Y. Q. et al. A plasmonic spanner for metal particle manipulation. Sci. Rep. 5, 15446 (2015).
 224.
Richardson, D. J., Fini, J. M. & Nelson, L. E. Spacedivision multiplexing in optical fibres. Nat. Photonics 7, 354–362 (2013).
 225.
Wang, J. Advances in communications using optical vortices. Photonics Res. 4, B14–B28 (2016).
 226.
Willner, A. E. et al. Optical communications using orbital angular momentum beams. Adv. Opt. Photonics 7, 66–106 (2015).
 227.
Lavery, M. P. J. et al. Freespace propagation of highdimensional structured optical fields in an urban environment. Sci. Adv. 3, e1700552 (2017).
 228.
Li, L. et al. Highcapacity freespace optical communications between a ground transmitter and a ground receiver via a UAV using multiplexing of multiple orbitalangularmomentum beams. Sci. Rep. 7, 17427 (2017).
 229.
Yan, Y. et al. Highcapacity millimetrewave communications with orbital angular momentum multiplexing. Nat. Commun. 5, 4876 (2014).
 230.
Jia, P. et al. Sidelobemodulated optical vortices for freespace communication. Opt. Lett. 38, 588–590 (2013).
 231.
Anguita, J. A., Herreros, J. & Cisternas, J. E. Generation and detection of multiple coaxial vortex beams for freespace optical communications. In Proc. Quantum Electronics and Laser Science Conference. (Optical Society of America, San Jose, California, United States, 2012).
 232.
Heng, X. B. et al. Allfiber stable orbital angular momentum beam generation and propagation. Opt. Express 26, 17429–17436 (2018).
 233.
Xie, Z. W. et al. Integrated (de)multiplexer for orbital angular momentum fiber communication. Photonics Res. 6, 743–749 (2018).
 234.
Lei, T. et al. Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings. Light Sci. Appl. 4, e257 (2015).
 235.
Ren, Y. X. et al. Orbital angular momentumbased space division multiplexing for highcapacity underwater optical communications. Sci. Rep. 6, 33306 (2016).
 236.
D’Ambrosio, V. et al. Complete experimental toolbox for alignmentfree quantum communication. Nat. Commun. 3, 961 (2012).
 237.
Zhu, L. et al. Orbital angular momentum mode groups multiplexing transmission over 2.6km conventional multimode fiber. Opt. Express 25, 25637–25645 (2017).
 238.
Mafu, M. et al. Higherdimensional orbitalangularmomentumbased quantum key distribution with mutually unbiased bases. Phys. Rev. A 88, 032305 (2013).
 239.
Sit, A. et al. Highdimensional intracity quantum cryptography with structured photons. Optica 4, 1006–1010 (2017).
 240.
Nicolas, A. et al. A quantum memory for orbital angular momentum photonic qubits. Nat. Photonics 8, 234–238 (2014).
 241.
Ding, D. S. et al. Quantum storage of orbital angular momentum entanglement in an atomic ensemble. Phys. Rev. Lett. 114, 050502 (2015).
 242.
Zhou, Z. Q. et al. Quantum storage of threedimensional orbitalangularmomentum entanglement in a crystal. Phys. Rev. Lett. 115, 070502 (2015).
 243.
Chen, L. X., Lei, J. J. & Romero, J. Quantum digital spiral imaging. Light Sci. Appl. 3, e153 (2014).
 244.
Nagali, E. et al. Optimal quantum cloning of orbital angular momentum photon qubits through Hong–Ou–Mandel coalescence. Nat. Photonics 3, 720–723 (2009).
 245.
Ndagano, B. et al. Creation and detection of vector vortex modes for classical and quantum communication. J. Lightwave Technol. 36, 292–301 (2018).
 246.
Wang, X. L. et al. Quantum teleportation of multiple degrees of freedom of a single photon. Nature 518, 516–519 (2015).
 247.
Wang, X. L. et al. 18qubit entanglement with six photons’ three degrees of freedom. Phys. Rev. Lett. 120, 260502 (2018).
 248.
Toninelli, E. et al. Concepts in quantum state tomography and classical implementation with intense light: a tutorial. Adv. Opt. Photonics 11, 67–134 (2019).
 249.
Sephton, B. et al. A versatile quantum walk resonator with bright classical light. PLoS ONE 14, e0214891 (2019).
 250.
Ndagano, B. et al. Characterizing quantum channels with nonseparable states of classical light. Nat. Phys. 13, 397–402 (2017).
 251.
Vieira, J. et al. Amplification and generation of ultraintense twisted laser pulses via stimulated Raman scattering. Nat. Commun. 7, 10371 (2016).
 252.
Lenzini, F. et al. Optical vortex interaction and generation via nonlinear wave mixing. Phys. Rev. A 84, 061801 (2011).
 253.
Jiang, W. et al. Computation of topological charges of optical vortices via nondegenerate fourwave mixing. Phys. Rev. A 74, 043811 (2006).
 254.
Li, G. X. et al. Continuous control of the nonlinearity phase for harmonic generations. Nat. Mater. 14, 607–612 (2015).
 255.
Tymchenko, M. et al. Gradient nonlinear PancharatnamBerry metasurfaces. Phys. Rev. Lett. 115, 207403 (2015).
 256.
KerenZur, S. et al. Nonlinear beam shaping with plasmonic metasurfaces. ACS Photonics 3, 117–123 (2015).
 257.
Li, G. X., Zentgraf, T. & Zhang, S. Rotational Doppler effect in nonlinear optics. Nat. Phys. 12, 736–740 (2016).
 258.
Musarra, G. et al. Rotationdependent nonlinear absorption of orbital angular momentum beams in ruby. Opt. Lett. 43, 3073–3075 (2018).
 259.
Qiu, C. W. & Yang, Y. J. Vortex generation reaches a new plateau. Science 357, 645 (2017).
 260.
Toda, Y. et al. Single orbital angular mode emission from externally feedbacked vertical cavity surface emitting laser. Appl. Phys. Lett. 111, 101102 (2017).
 261.
Miao, P. et al. Orbital angular momentum microlaser. Science 353, 464–467 (2016).
 262.
Wang, J. Metasurfaces enabling structured light manipulation: advances and perspectives [Invited]. Chin. Opt. Lett. 16, 050006 (2018).
 263.
Huang, L. L. et al. Dispersionless phase discontinuities for controlling light propagation. Nano Lett. 12, 5750–5755 (2012).
 264.
Sun, Y. Z. et al. Vector beam generation via micrometerscale photonic integrated circuits and plasmonic Nanoantennae. J. Opt. Soc. Am. B 33, 360–366 (2016).
 265.
Chen, P. et al. Digitalizing selfassembled chiral superstructures for optical vortex processing. Adv. Mater. 30, 1705865 (2018).
 266.
Chen, Y. et al. Mapping twisted light into and out of a photonic chip. Phys. Rev. Lett. 121, 233602 (2018).
 267.
Jin, Y. et al. Dynamic modulation of spatially structured polarization fields for realtime control of ultrafast lasermaterial interactions. Opt. Express 21, 25333–25343 (2013).
 268.
Allegre, O. J. et al. Complete wavefront and polarization control for ultrashortpulse laser microprocessing. Opt. Express 21, 21198–21207 (2013).
 269.
Toyoda, K. et al. Transfer of light helicity to nanostructures. Phys. Rev. Lett. 110, 143603 (2013).
 270.
Syubaev, S. et al. Direct laser printing of chiral plasmonic nanojets by vortex beams. Opt. Express 25, 10214–10223 (2017).
 271.
Masuda, K. et al. Azopolymer film twisted to form a helical surface relief by illumination with a circularly polarized Gaussian beam. Opt. Express 25, 12499–12507 (2017).
 272.
Takahashi, F. et al. Picosecond optical vortex pulse illumination forms a monocrystalline silicon needle. Sci. Rep. 6, 21738 (2016).
 273.
Torner, L., Torres, J. P. & Carrasco, S. Digital spiral imaging. Opt. Express 13, 873–881 (2005).
 274.
Tan, P. S. et al. Highresolution widefield standingwave surface plasmon resonance fluorescence microscopy with optical vortices. Appl. Phys. Lett. 97, 241109 (2010).
 275.
Zhang, C. L. et al. Perfect optical vortex enhanced surface plasmon excitation for plasmonic structured illumination microscopy imaging. Appl. Phys. Lett. 108, 201601 (2016).
 276.
Xie, X. S. et al. Harnessing the pointspread function for highresolution farfield optical microscopy. Phys. Rev. Lett. 113, 263901 (2014).
 277.
Wei, S. B. et al. Sub100nm resolution PSIM by utilizing modified optical vortices with fractional topological charges for precise phase shifting. Opt. Express 23, 30143–30148 (2015).
 278.
Kozawa, Y., Matsunaga, D. & Sato, S. Superresolution imaging via superoscillation focusing of a radially polarized beam. Optica 5, 86–92 (2018).
 279.
Willig, K. I. et al. STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis. Nature 440, 935–939 (2006).
 280.
Wang, S. B. & Chan, C. T. Lateral optical force on chiral particles near a surface. Nat. Commun. 5, 3307 (2014).
 281.
Brullot, W. et al. Resolving enantiomers using the optical angular momentum of twisted light. Sci. Adv. 2, e1501349 (2016).
 282.
Zhao, Y. et al. Chirality detection of enantiomers using twisted optical metamaterials. Nat. Commun. 8, 14180 (2017).
 283.
Jeffries, G. D. M. et al. Using polarizationshaped optical vortex traps for singlecell nanosurgery. Nano Lett. 7, 415–420 (2007).
 284.
Stellinga, D. et al. An organic vortex laser. ACS Nano 12, 2389–2394 (2018).
 285.
Lavery, M. P. J. et al. Detection of a spinning object using light’s orbital angular momentum. Science 341, 537–540 (2013).
 286.
Cvijetic, N. et al. Detecting lateral motion using light’s orbital angular momentum. Sci. Rep. 5, 15422 (2015).
 287.
Kravets, V. G. et al. Singular phase nanooptics in plasmonic metamaterials for labelfree singlemolecule detection. Nat. Mater. 12, 304–309 (2013).
 288.
Xie, G. D. et al. Using a complex optical orbitalangularmomentum spectrum to measure object parameters. Opt. Lett. 42, 4482–4485 (2017).
 289.
Fu, S. Y. & Gao, C. Q. Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams. Photonics Res. 4, B1–B4 (2016).
 290.
Li, Y., Yu, L. & Zhang, Y. X. Influence of anisotropic turbulence on the orbital angular momentum modes of HermiteGaussian vortex beam in the ocean. Opt. Express 25, 12203–12215 (2017).
 291.
Min, C. J. et al. Plasmonic nanoslits assisted polarization selective detour phase metahologram. Laser Photonics Rev. 10, 978–985 (2016).
 292.
Xie, Z. W. et al. Onchip spincontrolled orbital angular momentum directional coupling. J. Phys. D Appl. Phys. 51, 014002 (2017).
 293.
Zhang, C. & Ma, L. Detecting the orbital angular momentum of electromagnetic waves using virtual rotational antenna. Sci. Rep. 7, 4585 (2017).
 294.
Mei, S. T. et al. Onchip discrimination of orbital angular momentum of light with plasmonic nanoslits. Nanoscale 8, 2227–2233 (2016).
 295.
Foo, G., Palacios, D. M. & Swartzlander, G. A. Optical vortex coronagraph. Opt. Lett. 30, 3308–3310 (2005).
 296.
Lee, J. H. et al. Experimental verification of an optical vortex coronagraph. Phys. Rev. Lett. 97, 053901 (2006).
 297.
Swartzlander, G. A. et al. Astronomical demonstration of an optical vortex coronagraph. Opt. Express 16, 10200–10207 (2008).
 298.
Mawet, D. et al. Annular groove phase mask coronagraph. Astrophys. J. 633, 1191–1200 (2005).
 299.
Absil, O. et al. Three years of harvest with the vector vortex coronagraph in the thermal infrared. In Proc. SPIE 9908, Groundbased and Airborne Instrumentation for Astronomy VI. 99080Q (SPIE, Edinburgh, United Kingdom, 2016).
 300.
Aleksanyan, A., Kravets, N. & Brasselet, E. Multiplestar system adaptive vortex coronagraphy using a liquid crystal light valve. Phys. Rev. Lett. 118, 203902 (2017).
 301.
Aleksanyan, A. & Brasselet, E. Highcharge and multiplestar vortex coronagraphy from stacked vector vortex phase masks. Opt. Lett. 43, 383–386 (2018).
 302.
Tamburini, F. et al. Twisting of light around rotating black holes. Nat. Phys. 7, 195–197 (2011).
 303.
Battersby, S. Twisting the light away. New Sci. 182, 36–40 (2004).
 304.
Yin, J. Y. et al. Microwave vortexbeam emitter based on spoof surface plasmon polaritons. Laser Photonics Rev. 12, 1600316 (2018).
 305.
Marzo, A., Caleap, M. & Drinkwater, B. W. Acoustic virtual vortices with tunable orbital angular momentum for trapping of Mie particles. Phys. Rev. Lett. 120, 044301 (2018).
 306.
BialynickiBirula, I. & Charzyński, S. Trapping and guiding bodies by gravitational waves endowed with angular momentum. Phys. Rev. Lett. 121, 171101 (2018).
Acknowledgements
This work was funded by The National Key Research and Development Program of China (Grant No. 2017YFB1104500), Natural Science Foundation of Beijing Municipality (4172030), Beijing Young Talents Support Project (2017000020124G044), Leading talents of Guangdong province program (00201505), National Natural Science Foundation of China (U1701661, 91750205, 61975133, 11604218, 61975087), and Natural Science Foundation of Guangdong Province (2016A030312010, 2017A030313351). The first author Y.S. would like to thank Prof. Andrew Forbes at Wits University for useful discussions and Xilin Yang at the Beijing Institute of Technology for assistance with the graphics.
Author information
Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Shen, Y., Wang, X., Xie, Z. et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci Appl 8, 90 (2019). https://doi.org/10.1038/s4137701901942
Received:
Revised:
Accepted:
Published:
Further reading

Generation and manipulation of spiral beams in photovoltaic photorefractive crystal twinning with mirroring diffusion management
Optics Communications (2021)

Optical vortex knots and links via holographic metasurfaces
Advances in Physics: X (2021)

Small signal approximation method to determine output patterns of offaxis pumped solidstate lasers
Optics Communications (2021)

Tailoring diffraction of light carrying orbital angular momenta
Optics Letters (2020)

Axially controllable multiple orbital angular momentum beam generator
Applied Physics Letters (2020)