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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 Berry54 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 Stairs55, 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 well-known Möbius strip56. 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:
$$ \ell = \frac{1}{{2\pi }} \times {\oint}_C {\nabla \varphi \left( {\bf{r}} \right){\mathrm{dr}}} $$ (1) Fig. 2 Basic topological structure of vortex from art to science.
The topological structures of a the Penrose Stair55, b, c a Möbius strip56, and d the phase of a vortex soliton (Hilbert factor) are isomorphic, i.e., a physical value (displacement or angle) continually increases along a closed loop and coincides exactly with the origin after a roundtrip. c From ref. 56. Reprinted with permission from AAASwhere C is a tiny closed loop surrounding the singularity. For the light field with phase distribution exp(iℓθ) carrying OAM of ℓ$\hbar $ 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 non-integer TC was also experimentally and theoretically investigated in OVs57, 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 air53, water4, light1, electron59, and neutron60 vortex fields.
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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 quantum-classical-connection properties. There are already many review articles on OAM, especially on vortex generation61, 62, OAM on metasurfaces63, and basic OAM theories and applications64, 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 well-known Poynting picture to describe the AM of the photon66, 67, which leads to some difficulties, such as complex expressions of OAM and SAM, incompatibility with quantum optics, and the Abraham-Minkowski dilemma68. Here, we review the recently proposed new theory of the canonical picture69, 70, which can overcome these difficulties, to introduce basic properties of OAM. The canonical momentum of light is represented as
$$ {\bf{P}} = \frac{g}{2}{\rm Im} \left[ {\tilde \varepsilon {\bf{E}}^ \ast \cdot \left( \nabla \right){\bf{E}} + \tilde \mu {\bf{H}}^ \ast \cdot \left( \nabla \right){\bf{H}}} \right] $$ (2) 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
$$ {\bf{S}} = \frac{g}{2}{\rm Im} \left[ {\tilde \varepsilon {\bf{E}}^ \ast \times {\bf{E}} + \tilde \mu {\bf{H}}^ \ast \times {\bf{H}}} \right], {\bf{L}} = {\bf{r}} \;\times {\bf{P}} $$ (3) 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 z-axis:
$$ {\bf{E}}\left( {r, \theta , z} \right) = A\left( {r, z} \right)\frac{{{\hat{\bf x}} + m{\hat{\bf y}}}}{{\sqrt {1 + \left| m \right|^2} }}{\mathrm{exp}}\left( {ikz + i\ell \theta } \right) $$ (4) The average SAM and OAM can be derived as69, 70
$$ \frac{{\bf{S}}}{W} = \frac{\sigma }{\omega }\frac{{\bf{k}}}{k}, \frac{{\bf{L}}}{W} = \frac{\ell }{\omega }\frac{{\bf{k}}}{k} $$ (5) where the power density $W = \frac{{g\omega }}{2}\left({\tilde \varepsilon \left| {\bf{E}} \right|^2 + \tilde \mu \left| {\bf{H}} \right|^2} \right)$ 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 + $\hbar $ (-$\hbar $) per photon; the light with Hilbert factor exp(iℓθ) carries an OAM of ℓ$\hbar $ (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.
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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 state71. If the polarization state has a spatially varying vector distribution forming vortex-like 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 singularity72, 73. Based on the various topological disclinations of polarization, vector vortices can be categorized into many types, such as C-point, V-point, lemons, star, spider, and web, according to the actual vector morphology74. In contrast to the phase vortices carrying OAM, the vector vortices are always related to a complex SAM-OAM coupling; e.g., Fig. 4b shows a spider-like 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 space71.
Fig. 4 Formation of vector beam with space-polarization nonseparability.
a Circularly polarized OV with an azimuthally varying phase distribution. Such a state is considered separable, as it can be represented as the product of a spatially varying vortex phase and a polarization state vector. b Spider-like vector vortex represented as the superposition of the state of a with another state with the opposite phase variation and the opposite circular polarization. From ref. 71. Reprinted with permission from AAAS -
LG modes with circular symmetry are the earliest reported VBs carrying OAM16 and can be included in the general family of Hermite-Laguerre-Gaussian (HLG) modes with elliptical vortices75-77, 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:
$$ \begin{array}{c}{\mathrm{HLG}}_{n, m}\left( {\left. {{\bf{r}}, z} \right|\alpha } \right) = \frac{1}{{\sqrt {2^{N - 1}n!m!} }}\exp \left( { - \pi \frac{{\left| {\bf{r}} \right|^2}}{w}} \right){\mathrm{HL}}_{n, m}\left( {\left. {\frac{{\bf{r}}}{{\sqrt \pi w}}} \right|\alpha } \right)\\ \times \exp \left[ {{\mathrm{i}}kz + {\mathrm{i}}k\frac{{r^2}}{{2R}} - {\mathrm{i}}\left( {m + n + 1} \right)\vartheta } \right]\end{array} $$ (6) where HLn, m(·) is a Hermite-Laguerre (HL) polynomial75, ${r=(x, y)^{\mathrm{T}}} = {(r\, \cos{\varphi}, r\, \sin {\varphi})^{\mathrm{T}}}$, $R(z) = (z_R^2 + z^2)/z$, $kw^2(z) = 2(z_R^2 + z^2)/z_R$, $\vartheta (z) = \arctan (z/z_R)$, and zR is the Rayleigh range. For α = 0 or π∕2, the HLGn, m mode is reduced to the HGn, m or HGm, n mode. For α = π∕4 or 3π∕4, the HLGn, m mode is reduced to LGp, ±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 LGp, ℓ mode can be decomposed into a set of Hermite-Gaussian (HG) modes16, 17:
$$ {\mathrm{LG}}_{p, \pm \ell }\left( {x, y, z} \right) = \mathop {\sum}\limits_{K = 0}^{m + n} {({\pm{\mathrm{i}}})^Kb\left( {n, m, K} \right) \cdot } {\mathrm{HG}}_{m + n - K, K}\left( {x, y, z} \right) $$ (7) $$ b\left( {n, m, K} \right) = \left[ {\frac{{\left( {N - K} \right)!K!}}{{2^Nn!m!}}} \right]^{1/2}\frac{1}{{K!}}\left. {\frac{{{\mathrm{d}}^K}}{{{\mathrm{d}}t^K}}\left[ {\left( {1 - t} \right)^n\left( {1 + t} \right)^m} \right]} \right|_{t = 0} $$ (8) Fig. 5 Decomposition of LG vortex beams.
Examples of the decomposition of LG modes (LG0, 1 (a) and LG0, 2 (b)) into HG modes according to Eq. (7), where the insets in the dotted box show the corresponding vortex phase distributionswhich also interprets the transformation to an LGp, ℓ mode from an HGn, m mode through an astigmatic mode converter (AMC)17.
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The Ince-Gaussian (IG) mode78 is the eigenmode of the paraxial wave equation (PWE) separable in elliptical coordinates (ξ, η)79:
$$ \begin{array}{l}{\mathrm{IG}}_{u, v}^{e, o}\left( {\left. {x, y, z} \right|\epsilon } \right) = \\ \frac{{C^{e, o}}}{w}I_{u, v}^{e, o}\left( {{\mathrm{i}}\xi , \epsilon } \right)I_{u, v}^{e, o}\left( {\eta , \epsilon } \right)\exp \left( { - \frac{{x^{2}\ +\ y^{2}}}{{w^2}}} \right)\\ \exp \left[ {{\mathrm{i}}kz + {\mathrm{i}}k\frac{{x^{2}\ +\ y^2}}{{2R}} - {\mathrm{i}}\left( {u + 1} \right)\vartheta } \right]\end{array} $$ (9) where Ce, 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 multi-singularity array with OAM, named the helical-IG (HIG) modes80-82:
$$ {\mathrm{HIG}}_{u, v}^ \pm \left( {\left. {x, y, z} \right|\epsilon } \right) = {\mathrm{IG}}_{u, v}^e\left( {\left. {x, y, z} \right|\epsilon } \right) \pm {\mathrm{i}} \cdot {\mathrm{IG}}_{u, v}^o\left( {\left. {x, y, z} \right|\epsilon } \right) $$ (10) 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 is83
$$ \begin{array}{c}{\mathrm{SHEN}}_{n, m}\left( {x, y, z|\beta , \gamma } \right) = \mathop {\sum}\limits_{K = 0}^N {{\mathrm{e}}^{{\mathrm{i}}\beta K}b\left( {n, m, K} \right)} \\ \cdot \left\{ {\begin{array}{*{20}{c}} {\left( { - {\mathrm{i}}} \right)^K{\mathrm{IG}}_{N, N - K}^e\left( {x, y, z|\epsilon = 2{\mathrm{/}}\tan ^2\gamma } \right), {\mathrm{for}}\left( { - 1} \right)^K = 1} \\ {\left( { - {\mathrm{i}}} \right)^K{\mathrm{IG}}_{N, N - K + 1}^o\left( {x, y, z|\epsilon = 2{\mathrm{/}}\tan ^2\gamma } \right), {\mathrm{for}}\left( { - 1} \right)^K \ne\;1} \end{array}} \right.\end{array} $$ (11) 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 so-called SHEN sphere, to visualize the topological evolution of multi-singularity beams. Thus, the SHEN mode has great potential to characterize more general structure beams.
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Using the non-diffraction 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 as84
$$ {\mathrm{B}}_\ell \left( {r, \theta , z} \right) = J_\ell \left( {\mu r} \right)\exp \left( {{\mathrm{i}}\ell \theta } \right)\exp \left( {{\mathrm{i}}kz} \right) $$ (12) Bessel beams with ℓ ≠ 0 are VBs carrying ℓ$\hbar $ OAM. Another non-diffraction solution separable in elliptical coordinates is the Mathieu modes85,
$$ {\mathrm{M}}_m^e\left( {\left. {x, y, z} \right|\epsilon } \right) = C_{m}{\mathrm{Je}}_{m}\left( {\xi , \epsilon } \right){\mathrm{ce}}_{m}\left( {\eta , \epsilon } \right)\exp \left( {{\mathrm{i}}k_zz} \right) $$ (13) $$ {\mathrm{M}}_{m}^{o}\left( {\left. {x, y, z} \right|\epsilon } \right) = S_{m}{\mathrm{Jo}}_{m}\left( {\xi , \epsilon } \right){\mathrm{se}}_{m}\left( {\eta , \epsilon } \right)\exp \left( {{\mathrm{i}}k_zz} \right) $$ (14) where Cm and Sm are normalization constants, Jem and Jom are radial Mathieu functions, and cem and sem are angular Mathieu functions. Analogous to deriving the HIG mode, a helical Mathieu (HM) beam86 can carry multiple singularities and complex OAM87.
$$ {\mathrm{HM}}_m^ \pm \left( {\left. {x, y, z} \right|\epsilon } \right) = {\mathrm{M}}_m^e\left( {\left. {x, y, z} \right|\epsilon} \right) \pm {\mathrm{i}} \cdot {\mathrm{M}}_m^o\left( {\left. {x, y, z} \right|\epsilon} \right) $$ (15) High-order Bessel and HM beams are often called non-diffractive VBs, whose unique properties have been extended to a great number of applications, such as particle assembly and optical communication88, 89.
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When a resonator cavity fulfils the reentrant condition of a coupled quantum harmonic oscillator in SU(2) Lie algebra90, the laser mode undergoes frequency degeneracy with a photon performing as an SU(2) quantum coherent state coupled with a classical periodic trajectory91, 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 wave-packet function of a planar GM is given by92
$$ \Psi _{n_0}^M\left( {x, y, z;\phi _0\left| \Omega \right.} \right) = \frac{1}{{2^{M/2}}}\mathop {\sum}\limits_{K = 0}^M {\sqrt {\frac{{M!}}{{K!\left( {M - K} \right)!}}} }\\ \cdot {\mathrm{e}}^{{\mathrm{i}}K\phi _0} \cdot \psi _{n_0 + Q \cdot K, 0, s_0 - P \cdot K}^{\left( {{\mathrm{HG}}} \right)}\left( {x, y, z} \right) $$ (16) where phase ϕ0 is related to the classical periodic trajectory. $\psi _{n, m, s}^{\left({{\mathrm{HG}}} \right)}$ represents the HGn, m mode considering the frequency-dependent 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 obtained92:
$$ \Phi _{n_0}^M\left( {x, y, z;\phi _0\left| \Omega \right.} \right) = \frac{1}{{2^{M/2}}}\mathop {\sum}\limits_{K = 0}^M {\sqrt {\frac{{M!}}{{K!\left( {M - K} \right)!}}} } \\ \cdot {\mathrm{e}}^{{\mathrm{i}}K\phi _0} \cdot \psi _{0, \pm \left( {n_0 + Q \cdot K} \right), s_0 - P \cdot K}^{\left( {{\mathrm{LG}}} \right)}\left( {x, y, z} \right) $$ (17) where $\psi _{p, \ell, s}^{\left({{\mathrm{LG}}} \right)}$ represents the LGp, ℓ mode considering the frequency-dependent wavenumber. The vortex circular GM has many unique properties, such as an exotic 3D structure, multiple singularities, and fractional OAM92, 93. Note that there are other types of SU(2) modes related to OAM with special properties, such as Lissajous modes94, trochoidal modes95, polygonal VBs96, and SU(2) diffraction lattices97 as shown in Fig. 6d, e.
Fig. 6 Classical models of paraxial VBs.
a Evolution of the (Ⅰ) intensity and (Ⅱ) phase distributions of HLG modes as interposed states between HG and LG modes; b various (Ⅰ) intensity and (Ⅱ) phase distributions of IG and HLG modes82; c intensity distributions of a selection of (Ⅰ) odd and (Ⅱ) helical Mathieu beams88. (Ⅰ) Intensity and (Ⅱ) phase distributions of SU(2) vortex geometric modes for Ω = 1/4 (d) and Ω = 1/397 (e). SHEN spheres with orders of (n, m) = (3, 1) (f) and (n, m) = (0, 6) (g) along with represented mode (phase) fields at selected points83. b Reproduced from ref. 82, with the permission of AIP Publishing. c Reprinted with permission from ref. 88, OSA Publishing. d, e Reprinted with permission from ref. 97, OSA Publishing. f, g Reprinted with permission from ref. 83, OSA PublishingThe above forms are classical VBs in free space, which are just optical modes carrying OAM. In addition, there are OVs that are formed by non-OAM beams, as reviewed in the following.
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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 three-dimensional optical ellipse fields can be organized into an OMS, as theoretically proposed98, 99 and experimentally observed49. Currently, multitwist OMSs can be controlled in both paraxial and nonparaxial vector beams56, 100. By combining other spatial and optical parameters into OMSs, more complex structures, such as 3D solitons and topological knots, can be proposed for OVs101.
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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 field102. As a new form of OVs, vortex knots have stimulated many experimental observation and theoretical studies on the dynamics of knotted vortices102, 103. Vortex knots can also show many homologies, such as pigtail braid and Nodal trefoil knots104 as shown in Fig. 7c-f. Currently, researchers have realized the isolated manipulation and temporal control of optical vortex knots104, 105.
Fig. 7 Classical models of spatial nonparaxial OVs.
Polarization topology of optical Möbius strips with twisted TCs of-1/2 and-3/2 (a, b)56. Nodal trefoil knot and pigtail braid knot OVs (c, d) and corresponding phase distributions (e, f)104. Optical vortex knots of a threefold distorted loop (g), a trefoil knot (h), and a pair of linked rings (i)103. a, b From ref. 56. Reprinted with permission from AAAS. c-f Reprinted by permission from Nature Physics104, Copyright (2019). g-i Reprinted by permission from Nature Physics103, Copyright (2019)There are many other forms of OVs that cannot be fully covered in this paper. For instance, there are many free-space VB modes that carry OVs and OAM, such as elegant HLG beams106, Airy beams107, Pearcey beams108, and parabolic beams109. There are many morphologies of the non-beam spatial distribution of OVs with singularities fractality110. It is highly expected that many new formations of OVs will be reported and investigated in future explorations.
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The reflection of a VB generally does not satisfy the classical reflection law, i.e., the angle of incidence θi does not equal the angle of reflection θr. Instead, the reflected light has a spatial deflection effect related to the OAM of the VB111. The difference between θi and θr is related to the OAM of the beam, obeying the generalized law of reflection41
$$ \sin \left( {\theta _{\mathrm{r}}} \right) - \sin \left( {\theta _{\mathrm{i}}} \right) = \frac{\lambda }{{2\pi n}}\frac{{{\mathrm{d}}\phi }}{{{\mathrm{d}}x}} $$ (18) 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., ntsinθt ≠ nisinθi. The refraction is related not only to the angles of incidence and refraction (θi and θt) and the refractive indices but also to the OAM, obeying the generalized law of refraction41
$$ \sin \left( {\theta _{\mathrm{t}}} \right)n_{\mathrm{t}} - \sin \left( {\theta _{\mathrm{i}}} \right)n_{\mathrm{i}} = \frac{\lambda }{{2\pi }}\frac{{{\mathrm{d}}\phi }}{{{\mathrm{d}}x}} $$ (19) -
For conventional laser beams, the equal-inclination interference pattern is equispaced fringes, and the equal-thickness interference pattern is Newton's rings. However, for a VB, the pattern of equal-inclination 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 beam66. The equal-thickness 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 OAM112. The self-interference pattern can also show some bifurcation fringes112. These special interference fringes can be used in detection and measurement methods of vortices.
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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 triangular-aperture far-field diffraction of VBs, it has been used as an effective method for OAM detection and measurement of femtosecond vortices113, non-integer charge vortices114, and elliptical VBs115. Many other unique far-field diffraction patterns were investigated through a slit116, a square aperture117, a diamond-shaped aperture118, a circular aperture119, an off-axis circular aperture120, an isosceles right triangular aperture121, a sectorial screen122, and so on. The Fresnel diffraction of VBs was also studied123. Some special VBs, such as vector VBs124 and SU(2) VBs97, can even bring about special lattice structures in diffraction patterns. These special diffraction patterns can be used in vortex detection and measurement methods.
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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 applications74. To characterize a classical family of vector VBs, Holleczek et al. proposed a classical-quantum-connection model to represent cylindrically polarized beams on the Poincaré sphere125; this model was then extended to the high-order Poincaré sphere (HPS)126, which can reveal SAM-OAM conversion and more exotic vector beams, including radial and azimuthal polarization beams. In an experiment, controlled generation of HPS beams was realized127 as illustrated in Fig. 8f. As an improved formation of the HPS, the hybrid-order Poincaré sphere was theoretically proposed128, and the corresponding experimental controlled generation methods were also presented129, 130.
Fig. 8 Reflection, interference, diffraction, and polarization of VBs.
a Abnormal reflection of a VB111. b LG VBs with different TCs (first column) and corresponding interference patterns with a co-axis coherent planar wave (second column) and an inclined coherent planar wave (third column). Far-field diffraction patterns of VBs through a triangular aperture40 (c) and a single slit116 (d). e Near-field diffraction pattern of a VB123. f Polarization distribution of vector VBs on the HPS127. a Reprinted with permission from ref. 111. Copyright (2019) by the American Physical Society. c Reprinted with permission from ref. 40. Copyright (2019) by the American Physical Society. d Reprinted with permission from ref. 116, Copyright (2019), with permission from Elsevier. f Reprinted by permission from Nature Photonics127, Copyright (2019) -
Twisted photons31 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 ≥ $\hbar $/2131, 132. The general Fourier relationship between the angle and the OAM of twisted photons was also investigated133. In contrast to the polarization-entangled state with two dimensions, the OAM-entangled 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 realized47, 135. There are many other new quantum properties related to OAM beams, such as the spin-orbit interaction136-138, the Hanbury-Brown-Twiss effect139, quantum interference140, 141, and the spin Hall effect142, 143.
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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 detector144, the OAM sorter145, and spiral transformation146. For polarization vortices, the measurement should also consider the detection of the vector field. By introducing a space-variant structure into a half-wave plate to modulate the polarization, the TC of the polarization singularity in vector VBs can be measured147. 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 non-separability, SAM-OAM coupling, and vector factors of vector beams148, 149, which is widely applicable to more structured OVs.