Abstract

Besides a linear momentum, optical fields also carry angular momentum (AM), which has two intrinsic components: one is spin angular momentum related to the polarization state and the other is orbital angular momentum (OAM) caused by the helical phase due to the existence of a topological azimuthal charge. The two AM components of the optical field may not be independent of each other, especially if spin-to-orbit conversion (STOC) under high focusing creates a spin-dependent optical vortex in the longitudinal field. However, it would be very exciting to experimentally manifest and control the local OAM density. Here, we present a strategy for achieving STOC via a radial intensity gradient. The linearly varying radial phase provides an effective way to control the local AM density, which induces a counterintuitive orbital motion of the isotropic microparticles in optical tweezers without intrinsic OAM. Our work not only provides fundamental insights into the STOC of light, but could also have applications in optical micromanipulation.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Angular momentum (AM) is one of the important characteristics of light [1,2] and has attracted increasing attention in a variety of applications such as optical manipulations [3,4], quantum information [59], optical communications [10,11], and imaging [1214]. For paraxial fields in free space, AM can be formally separated into a spin angular momentum (SAM) associated with the right- or left-hand circular polarization corresponding to positive or negative helicity of $\sigma = + 1$ or $\sigma = - 1$, and an orbital angular momentum (OAM) caused by a helical phase of $\exp (jm\phi)$, where $m$ is the azimuthal topological charge and $\phi$ is the azimuthal angle. In general, SAM and OAM can be experimentally distinguished according to their different mechanical actions on microparticles. SAM usually makes the microparticle rotate around its own axis, while OAM causes orbital motion of the microparticles around the beam axis. This phenomenon is a conventional characterization method and distinguishing rule for SAM and OAM in experiments. In an extreme case, however, they cannot be separated. It has been known that spin-to-orbit conversion (STOC) could occur when a circularly polarized Gaussian beam is tightly focused by a high NA lens [15], generating a spin-induced optical vortex in the longitudinal field component. In fact, this conversion process verified in experiments depends mainly on the mechanical property of spin-dependent local AM [15] because the microparticles interact with part of the optical field.

For a circularly polarized optical field carrying a helical phase, the AM density in the direction of propagation is given by [16]

$${j_z} = {\varepsilon _0}\left({\omega m|u{|^2} - \frac{1}{2}\omega \sigma r\partial |u{|^2}/\partial r} \right),$$
where $u$ is a complex scalar function describing the distribution of the field amplitude, which satisfies the wave equation under the paraxial approximation. (See Section 1 of Supplement 1 for additional explanation.) Obviously, in Eq. (1), the first term originates from the contribution of intrinsic OAM caused by the topological defect introduced by the helical phase in the field; the second term, on the other hand, arises from the combination of SAM ($\sigma$) and the radial intensity gradient (RIG). We usually change the azimuthal topological charge $m$ to effectively control the AM carried by the beam. In fact, changing the RIG may also provide an effective way to manipulate the sign and value of the local AM density and it may help us to deeply understand the STOC. However, to the best of our knowledge, an analysis of the relation between the local AM density, STOC, and the RIG is still missing and remains a problem.

Here, we provide an effective way to manipulate the RIG and harness local AM density. We validate a spin-dependent local AM associated with the RIG rather than the spin-induced optical vortex. Such a spin-dependent local AM can induce a counterintuitive orbital motion of isotropic microparticles in optical tweezers. This breaks the cognition that the STOC will create a spin-dependent optical vortex in which the RIG plays an important role in the conversion process.

2. THEORETICAL ANALYSIS

To detect or characterize AM experimentally, an optical trap or tweezers provides an effective method because of the exchange of mechanical torque acting on microparticles based on the transfer of AM. Any practical axial-symmetry optical field always has countless axial-symmetry equal-intensity rings with nonzero RIG. As is well known, the dielectric microparticles cannot be trapped in these nonzero-RIG rings; instead, the dielectric microparticles can only be trapped in a “potential well,” which is the strongest intensity ring with zero RIG, because the microparticles experience the action of opposite light forces there. Therefore, the local AM density caused by the RIG can only be detected by probing the dielectric microparticles trapped in the strongest ring. Any finite-size trapped spherical microparticle will feel the opposite RIGs across the strongest ring. In general, the RIG in any plane in the propagation direction of light always exhibits a near-odd symmetry with respect to the strongest ring, like the Laguerre–Gaussian (LG) field with a doughnut intensity pattern, as shown in Fig. S5 in Supplement 1. This implies that the local AM density caused by the RIG has no almost contribution to the orbital motion of a spherical microparticle trapped in the strongest ring. When we use the dielectric microparticles to detect the AM density caused by the RIG, the key problem is how to achieve the nonsymmetric intensity distribution (or the non-odd symmetric intensity gradient) about the strongest ring.

The radial phase may provide an effective way to control the RIG. The radial phase can be classified into two categories: Category (i), which contains the odd power of radial coordinate only, and Category (ii), which contains the even power of radial coordinate only. Category (i) is completely different from Category (ii) in that the odd (even) power of the radial coordinate will (will not) dramatically change the spatial structure of the optical field. When the radial phase only contains the even power of the radial coordinate [17,18], a spin 3/2 light has been predicted [17]. Here, we attempt to harness the local AM density based on the RIG by using the linearly varying radial phase without the intrinsic OAM caused by the topological defect of helical phase, unlike [15], which requires a tightly focused circularly polarized vortex. Because the introduction of the linearly varying (odd power) radial phase results in a substantial change in the spatial structure of the optical field, for instance, a fundamental Gaussian beam will become a doughnut-like pattern.

Under the paraxial approximation, the transversal electric field of light with linearly varying radial phase without the helical phase in free space can be written as

$${{\boldsymbol E}_ \bot}(r) \propto A(r)\exp (j2q\pi r/{r_0})({\boldsymbol {\hat e}_x} + j\sigma {{\boldsymbol {\hat e}}_y}),$$
where $A(r)$ represents the amplitude of the optical field, $q$ is called the radial index that can be any number in the range $[- \infty , + \infty]$ and can describe the radial gradient of phase, ${r_0}$ is the radius of the field, and $\sigma$ represents the polarization state of light ($\sigma = \pm 1$ for right- and left-circularly polarized light and $\sigma = 0$ for linearly polarized). The linearly varying radial phase $\exp (j2q\pi r/{r_0})$ can be easily constructed by a spatial light modulator (SLM), which acts as an equivalent “axicon,” as shown in Figs. 1(a) and 1(b). If there is no “axicon” ($q = 0$), the input optical field will be focused in the Fourier plane (geometric focal plane) of the lens [$z = 0$, shown by the white dash-dot line in Figs. 1(a) and 1(b)]. In the presence of a linearly varying radial phase, however, the true focal fields will be moved toward the ${+}z$ (for $q \gt 0$) and ${-}z$ (for $q \lt 0$) directions away from the Fourier plane of the lens, as shown in Figs. 1(a) and 1(b), respectively. For more details, we should numerically simulate the transverse distribution of focused fields using the Richards–Wolf integral [19,20], as explained in Supplement 1. As shown in Figs. 1(c) and 1(d), the simulated $x - z$ plane intensity distributions in the vicinity of the Fourier plane of the lens with a focal length of $f = 500\;{\rm mm} $ indeed confirm the fact analyzed in Figs. 1(a) and 1(b). We also simulate the transverse intensity patterns of the focused field at the three different planes [➀ (red dashed line, $z = - 15\;{\rm mm} $), ➁ (white dash-dot line, $z = 0$) and ➂ (blue dashed line, $z = + 15\;{\rm mm} $) in Figs. 1(c) and 1(d)], which for both $q \gt 0$ ($q = + 10$) and $q \lt 0$ ($q = - 10$) exhibit the doughnut-like patterns similar to the focused vortex fields, as shown in Figs. 1(c1)–1(c3) and 1(d1)–1(d3). We define the radius of the strongest ring of the doughnut-like pattern as ${R_P}$. For greater clarity, as shown in Figs. 1(c1)–1(c3) and 1(d1)–1(d3), we have also simulated the radial-varying intensity (thick black solid curves), RIG (thin blue solid curves) and local AM density ${j_z}$ (red dot curves) in the three planes shown in Figs. 1(c) and 1(d) for the right-circularly polarized ($\sigma = + 1$) optical fields with $q = \pm 10$. One can see that the curves of ${j_z}$ and RIG have almost the same shape. In the Fourier plane of the lens, the doughnut-like patterns for both $q \gt 0$ and $q \lt 0$ have the same intensity pattern [Figs. 1(c2) and 1(d2)]; in particular, the radial intensity distribution across the strongest ring with the radius of ${R_P}$ is symmetric (i.e., the RIG is odd symmetric). In the $z \ne 0$ plane, however, the radial intensity distribution across the strongest ring is asymmetric. For instance, in the situation of $q \gt 0$, the doughnut pattern has the larger RIG within $r \lt {R_P}$ for $z \lt 0$ [Fig. 1(c1)] than within $r \gt {R_P}$ for $z \gt 0$ [Fig. 1(c3)]. The situation of $q \lt 0$ [Figs. 1(d1) and 1(d3)] is opposite to that of $q \gt 0$. This is completely different from the LG field, which always exhibits a near-symmetric intensity distribution, with respect to the strongest ring, in any plane along the propagation direction. Clearly, the linearly varying radial phase provides an effective method to control the RIG.
 figure: Fig. 1.

Fig. 1. Simulated transverse intensity patterns, radial-varying intensities, RIGs, and local AM densities of focused optical fields with the linearly varying radial phase for the right-hand circular polarization $\sigma = + 1$. (a)–(b) Schematic diagram of focusing process of optical fields with the radial index of $q \gt 0$ or $q \lt 0$. $z = 0$ represents the geometric focal plane (Fourier plane) of the lens. (c)–(d), Simulated $x - z$ plane intensity distribution of focused optical field with the radial index of $q = + 10$ or $q = - 10$ in the vicinity of the Fourier plane of the lens with a focal length of $f = 500\;{\rm mm} $ within the range of $z \in [- 30, + 30] {\rm mm}$. (c1)–(c3)–(d1)–(d3), Simulated transverse intensity patterns, radial-varying intensity (thick black solid curves), RIG (thin blue solid curves), and local AM density ${j_z}$ (red dot curves) at three different planes in (c)–d): ➀ $z = - 15\;{\rm mm} $, ➁ $z = 0$, and ➂ $z = + 15\;{\rm mm} $. All the pictures have the same size of $1.76 \times 1.76\;{{\rm mm}^2}$ in (c1)–(c3) and (d1)–(d3).

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If the size of microparticles is comparable to the width of the doughnut pattern of the focused optical field with linearly varying radial phase, we consider the integral of the local AM density ${j_z}$ across the strongest ring (from ${R_P} - a$ to ${R_P} + a$) to describe the local interaction between the focused field and microparticles, as further explained in Section 2 of Supplement 1:

$$\begin{split}{J_z^L}&{\propto \int_{{R_P} - a}^{{R_P} + a} {j_z}dr = \int_{{R_P} - a}^{{R_P} + a} \omega r\sigma \frac{{\partial |A(r{{)|}^2}}}{{\partial r}}{\rm d}r}\\&{= \omega \sigma \left[{\int_{{R_P} - a}^{{R_P} + a} r|A(r{{)|}^2}{\rm d}r - \frac{1}{2}{r^2}\left. {|A(r{{)|}^2}} \right|_{{R_P} - a}^{{R_P} + a}} \right].}\end{split}$$

The gradient force constrains the microparticles to the strongest ring and $a$ is the radius of the probing microparticle. As shown in Fig. 1, in the Fourier plane of the lens, the local doughnut-like pattern has a Gaussian-like intensity distribution as $\exp [(r - {R_P}{)^2}/{a^2}]$, so the net local integral AM $J_z^L$ will be near zero, which cannot result in the orbital motion of microparticles. In the plane $z \ne 0$; however, $J_z^L$ will be nonzero due to the asymmetric radial intensity (i.e., non-odd symmetric RIG) about $r = {R_P}$, meaning that the trapped microparticles will move along a ring orbit. If we choose the diameter of microparticles to be $30\,\,\unicode{x00B5}{\rm m}$, the simulated local integral AM values are $J_z^L = - 0.57$ for $q = + 10$ and $J_z^L = + 0.54$ for $q = - 10$, in the plane of $z = + 15\;{\rm mm} $, respectively. Obviously, the microparticles will move along the ring orbit although the input optical field has only SAM without the intrinsic OAM caused by the helical phase. The sense of the orbital motion depends on the sign of the radial index $q$ and the handedness $\sigma$ of SAM. For given $q$ and $\sigma$, $J_z^L$ also depends on the longitudinal coordinate $z$; its sign will change across the Fourier plane of the lens, which is completely different from the intrinsic OAM causes by the helical phase. Of course, $J_z^L$ also depends on SAM. When the optical field is linearly polarized (i.e., $\sigma = 0$), the microparticles have no orbital motion no matter what the intensity distribution of the focused field. T herefore, for nonzero $J_z^L$, the SAM and the linearly varying radial phase are both indispensable.

 figure: Fig. 2.

Fig. 2. Schematic diagrams of emergence of the azimuthal force induced by the intensity gradient and circular polarization. The green color shows the intensity distribution of the annular focus ring: (a) symmetric and (b) asymmetric. Blue circles show the circulation cells of the spin flow at different position and their size is proportional to the local intensity. The red arrows show the net force by uncompleted compensation of the adjacent circulation cells. The open red arrow shows the azimuthal resultant force if we consider the microparticle as a rigid body.

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We can also understand the physical process above by the simple phenomenological model of the spin flow schematically illustrated in Fig. 2. The physical ground is the energy circulation in the circularly polarized light [21]. We can imagine what the rotation of the field vectors takes place in “every point” of the focused field and the energy circulates within microscopic “cells.” For the circularly polarized light with uniform intensity, all the “cells” are identical, the contributions of spin flow from the adjacent cells counterbalance and the macroscopic energy flow is absent. The compensation is not complete if the spin flow, which is shown by blue circles with arrows in Fig. 2 (the larger and thicker blue circles with arrows indicate the stronger spin flow), of the adjacent cells are different (i.e., the intensity is transversely inhomogeneous), and the net spin flow between adjacent cells (red arrows in Fig. 2; the thicker red arrows indicate the stronger net spin flow) is along the azimuth (i.e., orthogonal to the radial intensity gradient). In the $z = 0$ plane, the doughnut-like field exhibits the Gaussian-like symmetric intensity distribution $\exp [(r - {R_P}{)^2}/{a^2}]$ about the strongest ring with the radius of ${R_P}$, as shown by the symmetric dark green doughnut-like in Fig. 2(a); that is, the RIG is odd symmetric about $r = {R_P}$, so the spin flow is also odd symmetric about $r = {R_P}$. Therefore, the macroscopic spin flow vanishes, and this is the reason why a trapped calcite microparticle only spins about own axis while has no orbital motion, which is similar to the circular polarized LG beam [22]. In contrast, in the $z \ne 0$ plane, the optical field becomes asymmetric in intensity about $r = {R_P}$, as shown by the asymmetric dark green doughnut-like in Fig. 2(b); that is, the RIG lacks the odd symmetry about $r = {R_P}$, so the spin flow also has no odd symmetry about $r = {R_P}$. As a result, the macroscopic spin flow becomes nonzero along the azimuth [shown by the open red arrow in Fig. 2(b)], which is the reason why the trapped microparticles have the orbital motion.

3. EXPERIMENTAL RESULTS

To confirm the theoretical analysis and simulation results above, we experimentally generate the optical fields with a linearly varying radial phase, using a spatial light modulator (SLM) to load a hologram of blazed grating, as shown in Fig. S10 in Supplement 1. Under the focusing condition of $f = 500\;{\rm mm} $ and ${\rm NA} = {0.01}$, we experimentally measure the transverse intensity patterns of focused fields of the right-handed circularly polarized optical fields with radial indices of $q = \pm 10$ by CCD, in three different planes ($z = - 15\;{\rm mm} $, $z = 0$, $z = + 15\;{\rm mm} $), as shown by the insets in Figs. 3(a)–3(f). For clarity, the blue solid curves in Figs. 3(a)–3(f) also plot the radial-varying intensities read from the images recorded by CCD. Correspondingly, the red dot curves in Figs. 3(a)–3(f) plot the local AM densities calculated from the radial-varying intensities shown by the blue solid curves in Figs. 3(a)–3(f). Obviously, the experimental results in Fig. 3 are in good agreement with the simulated doughnut-like patterns in Fig. 1.

 figure: Fig. 3.

Fig. 3. Experimentally measured radial-varying intensity (blue solid curves) and corresponding calculated local AM density (red dot curves) of the focused field by the lens ($f = 500\;{\rm mm} $ and NA = 0.01) in three different planes ($z = - 15\;{\rm mm} $, $z = 0$, $z = + 15\;{\rm mm} $). (a)–(c) Radial index $q = + 10$. (d)–(f) Radial index $q = - 10$. The light is right-handed circularly polarized ($\sigma = + 1$). Insets: transverse intensity patterns measured by CCD.

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Optical tweezers are a useful tool that has been successfully used to probe the AM flow by means of the interaction of microparticles with light [23,24]. Here, we also perform the optical tweezers experiment to validate the local AM property of the optical fields with linearly varying radial phase, as shown in Fig. S10 in Supplement 1. We are interested in realizing the counterintuitive orbital motion of the microparticles caused by the linearly varying radial phase; hence, we choose neutral isotropic colloidal microspheres with a radius of $1.6\,\,\unicode{x00B5}{\rm m}$ as probing microparticles in an experiment. To observe the obvious orbital motion of the trapped microparticles, we use an optical tweezers system composed of an objective (with a high NA, NA = 0.75) by an incident optical field with a power of 108 mW. We also simulate the transverse intensity pattern, radial-varying intensity, and local AM density of the focused light fields with different radial index and SAM, as shown in Figs. S1–S4 of Supplement 1. They are very similar to the weak focusing situations mentioned above (Figs. 1 and 3). For the case of $q = + 6$, the close-packed nine microparticles are trapped in the doughnut-like field behind the geometric focal plane of the objective ($z \gt 0$), and exhibit clockwise or counterclockwise orbital motion, depending on the handedness of SAM, as shown in Figs. 4(a) and 4(b). When the light becomes linearly polarized ($\sigma = 0$), no orbital motion of microparticles is observed. The maximum number of microparticles trapped in the doughnut-like field can be controlled by changing the radial index $q$. For the case of $q = + 7$ in Fig. 4(c), the trapped microparticles have increased to 10 and move clockwise for the right-circularly polarized light; however, when switching to $q = - 7$ in Fig. 4(d), the motion direction of the trapped 10 microparticles is synchronously reversed. Obviously, the orbital motion of the trapped microparticles arises from the transfer of the local AM $J_z^L$ to microparticles through the RIG, which can be effectively controlled by the radial phase.

 figure: Fig. 4.

Fig. 4. Camera snapshots of continuously rotating microparticles trapped by the focused light with the linearly varying radial phase behind the focal plane ($z \gt 0$) of the objective (NA = 0.75). (a) $q = + 6$ and $\sigma = + 1$; (b) $q = + 6$ and $\sigma = - 1$; (c) $q = + 7$ and $\sigma = + 1$; and (d) $q = - 7$ and $\sigma = + 1$. The red arrows denote the orbital rotational direction, and the corresponding elapsing time $t$ is given in each snapshot.

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In addition, $J_z^L$ also has an important feature: its longitudinal dependence. The trapped microparticles move clockwise for $q = + 6$ and $\sigma = + 1$ in the case of $z \lt 0$ in Fig. 5(a), while the sense of the orbital motion of the trapped microparticles is reversed for $q = + 9$ and $\sigma = + 1$ in the case of $z \gt 0$ in Fig. 5(b). We can load the hologram with the linearly varying radial phase of $q = + 6$ and vortex phase of $m = + 1$ on SLM simultaneously, then the focused field still exhibits the doughnut shape. When $\sigma = + 1$, the trapped microparticles almost stop behind the geometric focal plane ($z \gt 0$), and the local AM from the linearly varying radial phase is compensated with the intrinsic OAM of helical phase, which means that the net $J_z^L$ acting on microparticles is almost zero in Fig. 6(a). Switching the polarization from $\sigma = + 1$ to $\sigma = - 1$, the trapped microparticles move clockwise with a shorter orbital period of 8 s in Fig. 6(b). Clearly, $J_z^L$ caused by the RIG of spin optical field can have the same or opposite sign as the intrinsic OAM originated from the vortex phase, as shown in Fig. S6 in Supplement 1.

 figure: Fig. 5.

Fig. 5. Camera snapshots of continuously rotating microparticles trapped by the focused light with the linearly varying radial phase and circular polarization $\sigma = + 1$ for different positions: (a) before the focal plane ($z \lt 0$), $q = + 6$ and (b) behind the focal plane ($z \gt 0$), $q = + 9$. The objective has a NA = 0.75. The red arrows indicate the orbital rotational direction, and the corresponding elapsing time $t$ is given in each snapshot.

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 figure: Fig. 6.

Fig. 6. Camera snapshots of continuously rotating microparticles trapped by the focused light with the azimuthal and radial phase behind the focal plane ($z \gt 0$) of the objective (NA = 0.75) for different spins: (a) $\sigma = + 1$ and (b) $\sigma = - 1$. The topological charge $m = + 1$, radial index $q = + 6$. The red arrows denote the orbital rotational direction, and the corresponding elapsing time $t$ is given in each snapshot.

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4. DISCUSSION AND CONCLUSION

We experimentally demonstrate the fact that, in the absence of intrinsic OAM, the net nonzero local AM density may not only be different in magnitude but also has a different sign from the SAM in the linearly varying radial phase. The sense and velocity of the orbital motion of microparticles depend on SAM and the RIG, and this local AM can be continuously changed by selecting arbitrary radial index $q$. The effective torque on microparticles from the local AM $J_z^L$ is equivalent to that from the intrinsic OAM; in this sense, the radial phase provides a new way to perform optical manipulation. Our scheme offers continuous control of the local AM density based on the linearly varying radial phase and is also compatible with other degrees of freedom of light. The observed orbital motion of the isotropic microparticles in an optical tweezers system using the optical fields without intrinsic OAM may show the same the experimental phenomena as the OAM caused by STOC from a tightly focused circularly polarized Gaussian beam [15] and an inhomogeneous and anisotropic metamaterial [25]. By numerical simulation, the intensity of the longitudinal field is very small compared to the transverse fields, as show in Fig. S8 in Supplement 1. It implies that the RIG plays an important role in the STOC, as shown in Fig. S9 in Supplement 1.

More importantly, the spin-orbital coupling originates from the helical phase front of the longitudinal component produced by the strongly focused circularly polarized light. In particular, the sense of the OAM carried by the longitudinal component only depends on the spin handedness of the incident light and is independent from the longitudinal propagation distance. This is quite different from STOC caused by the RIG in our work. As a result, the observed orbital motion of the microparticles under the strongly focusing condition arises from STOC associated with the RIG rather than the spin-orbital coupling produced by the strongly focused circularly polarized field. We believe that the contribution of the latter to the orbital motion of microparticles we observed can be ignored.

We have theoretically predicted and verified STOC associated with the RIG through optical trapping experiments. We believe this result breaks the limitation that orbital motion must be associated with the azimuthal phase gradient and enriches the category of AM. This spin-dependent AM also offers a deeper understanding of SAM in optics as well as the conversion of spin to orbit. Since this AM can be easily generated and arbitrarily tunable, it opens what we believe, to the best of our knowledge, is a novel route to control light–matter interaction and optical metrology and it could have applications in optical micromanipulation and microfabrication. In addition, it is also expected to be found in other natural waves such as electron beams and acoustic waves [2628].

Funding

National Natural Science Foundation of China (11774183, 11922406, 12074197); National Key Research and Development Program of China (2017YFA0303700, 2017YFA0303800, 2019YFA0308700, 2020YFA0309500).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

1. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008). [CrossRef]  

2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]  

3. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997). [CrossRef]  

4. D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017). [CrossRef]  

5. M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018). [CrossRef]  

6. M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016). [CrossRef]  

7. J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020). [CrossRef]  

8. J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017). [CrossRef]  

9. G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014). [CrossRef]  

10. J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4, B14–B28 (2016). [CrossRef]  

11. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015). [CrossRef]  

12. L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014). [CrossRef]  

13. X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020). [CrossRef]  

14. H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020). [CrossRef]  

15. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007). [CrossRef]  

16. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef]  

17. O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018). [CrossRef]  

18. Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020). [CrossRef]  

19. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959). [CrossRef]  

20. X. Z. Gao, Y. Pan, G. L. Zhang, M. D. Zhao, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang, “Redistributing the energy flow of tightly focused ellipticity-variant vector optical fields,” Photon. Res. 5, 640–648 (2017). [CrossRef]  

21. A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011). [CrossRef]  

22. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002). [CrossRef]  

23. M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011). [CrossRef]  

24. X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010). [CrossRef]  

25. M. Kang, J. Chen, X. L. Wang, and H. T. Wang, “Twisted vector field from an inhomogeneous and anisotropic metamaterial,” J. Opt. Soc. Am. B 29, 572–576 (2012). [CrossRef]  

26. S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017). [CrossRef]  

27. A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018). [CrossRef]  

28. X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016). [CrossRef]  

References

  • View by:

  1. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
    [Crossref]
  2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
    [Crossref]
  3. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
    [Crossref]
  4. D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
    [Crossref]
  5. M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
    [Crossref]
  6. M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
    [Crossref]
  7. J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
    [Crossref]
  8. J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
    [Crossref]
  9. G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
    [Crossref]
  10. J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4, B14–B28 (2016).
    [Crossref]
  11. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
    [Crossref]
  12. L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
    [Crossref]
  13. X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020).
    [Crossref]
  14. H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
    [Crossref]
  15. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
    [Crossref]
  16. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [Crossref]
  17. O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018).
    [Crossref]
  18. Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
    [Crossref]
  19. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
    [Crossref]
  20. X. Z. Gao, Y. Pan, G. L. Zhang, M. D. Zhao, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang, “Redistributing the energy flow of tightly focused ellipticity-variant vector optical fields,” Photon. Res. 5, 640–648 (2017).
    [Crossref]
  21. A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
    [Crossref]
  22. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
    [Crossref]
  23. M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
    [Crossref]
  24. X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
    [Crossref]
  25. M. Kang, J. Chen, X. L. Wang, and H. T. Wang, “Twisted vector field from an inhomogeneous and anisotropic metamaterial,” J. Opt. Soc. Am. B 29, 572–576 (2012).
    [Crossref]
  26. S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
    [Crossref]
  27. A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
    [Crossref]
  28. X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
    [Crossref]

2020 (4)

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020).
[Crossref]

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

2018 (3)

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018).
[Crossref]

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

2017 (4)

X. Z. Gao, Y. Pan, G. L. Zhang, M. D. Zhao, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang, “Redistributing the energy flow of tightly focused ellipticity-variant vector optical fields,” Photon. Res. 5, 640–648 (2017).
[Crossref]

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

2016 (3)

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4, B14–B28 (2016).
[Crossref]

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

2015 (1)

2014 (2)

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

2012 (1)

2011 (3)

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

2010 (1)

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

2008 (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

2007 (1)

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

2002 (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

1997 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Ahmed, N.

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Ashrafi, N.

Ashrafi, S.

Babiker, M.

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Bao, C.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Bekshaev, A.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

Bliokh, K. Y.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

Bowman, R.

M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

Burge, R. E.

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

Bürger, J.

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

Cai, W. Q.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Cao, Y.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Chen, J.

M. Kang, J. Chen, X. L. Wang, and H. T. Wang, “Twisted vector field from an inhomogeneous and anisotropic metamaterial,” J. Opt. Soc. Am. B 29, 572–576 (2012).
[Crossref]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Chen, L.

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
[Crossref]

Chen, Y. A.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Cheng, J. C.

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

D’Ambrosio, V.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Deng, L.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Dholakia, K.

Ding, J.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Ding, W.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Ding, X.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Ekert, A. K.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

El Gawhary, O.

O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018).
[Crossref]

Erhard, M.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Fang, X.

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020).
[Crossref]

Fickler, R.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

Gao, D.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Gao, X. Z.

Gu, M.

X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020).
[Crossref]

Gu, Y.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Guo, C.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Guo, C. S.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Guo, F.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Han, X.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

He, Z. P.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Huang, H.

Huang, T. J.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Huang, Y. M.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Huber, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Jang, J.

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Jiang, X.

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

Kang, M.

Krenn, M.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Kuang, Y. W.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Lata, J.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Lavery, M. P. J.

Lei, J.

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
[Crossref]

Li, J.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Li, L.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Li, M. Y.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Li, P.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Li, S. L.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Li, Y.

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Li, Y. H.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Li, Y. N.

Liang, B.

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

Liao, S. K.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Lim, C. T.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Liu, D. Q.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Liu, J. Y. W. Y.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Liu, L.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Liu, N. L.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Liu, W. Y.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Lloyd, S. M.

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Lu, C. Y.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Maier, S. A.

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

Malik, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Man, Z.

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

Marrucci, L.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Molisch, A. F.

Nieto-Vesperinas, M.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Ozcelik, A.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

Padgett, M. J.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[Crossref]

Pan, J. W.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Pan, Y.

Peng, C. Z.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Qiu, C. W.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Rahman, M.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Ramachandran, S.

Ren, H.

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020).
[Crossref]

Ren, J. G.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Ren, Y.

Ren, Z. C.

Rho, J.

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Romero, J.

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
[Crossref]

Rufo, J.

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Sciarrino, F.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Shang, P.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Shu, R.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Simpson, N. B.

Slussarenko, S.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Soskin, M.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

Sponselli, A.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Thirunavukkarasu, G.

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Tian, K.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Tu, C. H.

Tur, M.

Urbach, H. P.

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018).
[Crossref]

Vallone, G.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Van Mechelen, T.

O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018).
[Crossref]

Villoresi, P.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Wan, S.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Wang, H. T.

Wang, J.

Wang, J. Y.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Wang, X. B.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Wang, X. L.

M. Kang, J. Chen, X. L. Wang, and H. T. Wang, “Twisted vector field from an inhomogeneous and anisotropic metamaterial,” J. Opt. Soc. Am. B 29, 572–576 (2012).
[Crossref]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Willner, A. E.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Wu, H. Y.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Xi, Z.

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

Xie, G.

Xu, F.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Xu, P.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Yan, Y.

Yang, K. X.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Yang, M.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Yao, A. M.

Yao, Y. Q.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Yin, J.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Yong, H. L.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Yuan, J.

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Yuan, X.

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

Zeilinger, A.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Zhang, G. L.

Zhang, L.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

Zhang, Q.

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

Zhang, T.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

Zhao, M. D.

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Zhao, Z.

Zheng, R. H.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Zhu, Z. C.

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Adv. Opt. Photon. (2)

J. Opt. (1)

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[Crossref]

Light Sci. Appl. (3)

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6, e17039 (2017).
[Crossref]

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3, e153 (2014).
[Crossref]

Nat. Methods (1)

A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, “Acoustic tweezers for the life sciences,” Nat. Methods 15, 1021 (2018).
[Crossref]

Nat. Nanotechnol. (1)

H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15, 948–955 (2020).
[Crossref]

Nat. Photonics (3)

X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14, 102–108 (2020).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[Crossref]

Nature (2)

J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, R. Shu, Y. M. Huang, L. Deng, L. Li, Q. Zhang, N. L. Liu, Y. A. Chen, C. Y. Lu, X. B. Wang, F. Xu, J. Y. Wang, C. Z. Peng, A. K. Ekert, and J. W. Pan, “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature 582, 501–505 (2020).
[Crossref]

J. G. Ren, P. Xu, H. L. Yong, L. Zhang, S. K. Liao, J. Y. W. Y. Liu, W. Q. Cai, M. Y. Li, K. X. Yang, X. Han, Y. Q. Yao, J. Li, H. Y. Wu, S. Wan, L. Liu, D. Q. Liu, Y. W. Kuang, Z. P. He, P. Shang, C. Guo, R. H. Zheng, K. Tian, Z. C. Zhu, N. L. Liu, C. Y. Lu, R. Shu, Y. A. Chen, C. Z. Peng, J. Y. Wang, and and J. W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549,70–73 (2017).
[Crossref]

Opt. Lett. (1)

Photon. Res. (2)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Phys. Rev. Lett. (7)

O. El Gawhary, T. Van Mechelen, and H. P. Urbach, “Role of radial charges on the angular momentum of electromagnetic fields: spin-3/2 light,” Phys. Rev. Lett. 121, 123202 (2018).
[Crossref]

Z. Man, Z. Xi, X. Yuan, R. E. Burge, and H. P. Urbach, “Dual coaxial longitudinal polarization vortex structures,” Phys. Rev. Lett. 124, 103901 (2020).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. Zhang, “Convert acoustic resonances to orbital angular momentum,” Phys. Rev. Lett. 117, 034301 (2016).
[Crossref]

Proc. R. Soc. London A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Rev. Mod. Phys. (1)

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Supplementary Material (1)

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Supplement 1       Supplementary Information

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Simulated transverse intensity patterns, radial-varying intensities, RIGs, and local AM densities of focused optical fields with the linearly varying radial phase for the right-hand circular polarization $\sigma = + 1$. (a)–(b) Schematic diagram of focusing process of optical fields with the radial index of $q \gt 0$ or $q \lt 0$. $z = 0$ represents the geometric focal plane (Fourier plane) of the lens. (c)–(d), Simulated $x - z$ plane intensity distribution of focused optical field with the radial index of $q = + 10$ or $q = - 10$ in the vicinity of the Fourier plane of the lens with a focal length of $f = 500\;{\rm mm} $ within the range of $z \in [- 30, + 30] {\rm mm}$. (c1)–(c3)–(d1)–(d3), Simulated transverse intensity patterns, radial-varying intensity (thick black solid curves), RIG (thin blue solid curves), and local AM density ${j_z}$ (red dot curves) at three different planes in (c)–d): ➀ $z = - 15\;{\rm mm} $, ➁ $z = 0$, and ➂ $z = + 15\;{\rm mm} $. All the pictures have the same size of $1.76 \times 1.76\;{{\rm mm}^2}$ in (c1)–(c3) and (d1)–(d3).
Fig. 2.
Fig. 2. Schematic diagrams of emergence of the azimuthal force induced by the intensity gradient and circular polarization. The green color shows the intensity distribution of the annular focus ring: (a) symmetric and (b) asymmetric. Blue circles show the circulation cells of the spin flow at different position and their size is proportional to the local intensity. The red arrows show the net force by uncompleted compensation of the adjacent circulation cells. The open red arrow shows the azimuthal resultant force if we consider the microparticle as a rigid body.
Fig. 3.
Fig. 3. Experimentally measured radial-varying intensity (blue solid curves) and corresponding calculated local AM density (red dot curves) of the focused field by the lens ($f = 500\;{\rm mm} $ and NA = 0.01) in three different planes ($z = - 15\;{\rm mm} $, $z = 0$, $z = + 15\;{\rm mm} $). (a)–(c) Radial index $q = + 10$. (d)–(f) Radial index $q = - 10$. The light is right-handed circularly polarized ($\sigma = + 1$). Insets: transverse intensity patterns measured by CCD.
Fig. 4.
Fig. 4. Camera snapshots of continuously rotating microparticles trapped by the focused light with the linearly varying radial phase behind the focal plane ($z \gt 0$) of the objective (NA = 0.75). (a) $q = + 6$ and $\sigma = + 1$; (b) $q = + 6$ and $\sigma = - 1$; (c) $q = + 7$ and $\sigma = + 1$; and (d) $q = - 7$ and $\sigma = + 1$. The red arrows denote the orbital rotational direction, and the corresponding elapsing time $t$ is given in each snapshot.
Fig. 5.
Fig. 5. Camera snapshots of continuously rotating microparticles trapped by the focused light with the linearly varying radial phase and circular polarization $\sigma = + 1$ for different positions: (a) before the focal plane ($z \lt 0$), $q = + 6$ and (b) behind the focal plane ($z \gt 0$), $q = + 9$. The objective has a NA = 0.75. The red arrows indicate the orbital rotational direction, and the corresponding elapsing time $t$ is given in each snapshot.
Fig. 6.
Fig. 6. Camera snapshots of continuously rotating microparticles trapped by the focused light with the azimuthal and radial phase behind the focal plane ($z \gt 0$) of the objective (NA = 0.75) for different spins: (a) $\sigma = + 1$ and (b) $\sigma = - 1$. The topological charge $m = + 1$, radial index $q = + 6$. The red arrows denote the orbital rotational direction, and the corresponding elapsing time $t$ is given in each snapshot.

Equations (3)

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j z = ε 0 ( ω m | u | 2 1 2 ω σ r | u | 2 / r ) ,
E ( r ) A ( r ) exp ( j 2 q π r / r 0 ) ( e ^ x + j σ e ^ y ) ,
J z L R P a R P + a j z d r = R P a R P + a ω r σ | A ( r ) | 2 r d r = ω σ [ R P a R P + a r | A ( r ) | 2 d r 1 2 r 2 | A ( r ) | 2 | R P a R P + a ] .

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