Open Access
Add to BibSonomy
Abstract
Perfect vortex (PV) beam has seen significant advances in fields like particle manipulation, optical tweezers, and particle trapping, due to the fact that its ring radius is independent of the topological charge. Although geometric-phase metasurfaces have been proposed to generate PV beams, it always relies on circularly or elliptically polarized incident light, which hinders the miniaturization of compact optical devices. Here, using orthogonal decomposition of polarization vectors (ODPV), we proposed a geometric-phase metasurface, which breaks the dependence of circular polarization, to generate PV beam. In the design of the metasurface, we introduced PV phase profiles corresponding to the left-handed circularly polarized (LCP) component and the right-handed circularly polarized (RCP) component into the metasurface based on the principle of ODPV. We further determined the rotation angle of each nanostructure of the metasurface by calculating the argument of the composite vector of LCP and RCP in the transmission field. Simulation results show that the proposed geometric-phase metasurface can generate the PV beam upon the illumination of a linearly polarized incident. Moreover, the PV beam with polarization-rotated functionality is achieved by setting the polarization rotation angle. Furthermore, dual PV beams with orthogonal polarization states is realized at the same time by superimposing two sets of phase profiles on a single metasurface. It is also demonstrated that the PV beam parameters, such as ring radius and/or topological charge, can be set on demand in the metasurface design. The proposed metasurface has the exceptional advantage of high fabrication tolerance and is optical path miniaturization friendly, and will open a new avenue in advanced compact and integrated optical systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Possessing orbital angular momentum (OAM) associated with helical phase-front, vortex beams [1] have been paid increasing attention in many pivotal fields such as optical trapping [2–4], optical communication [5–7] and quantum information processing [8–10]. However, the annular intensity profiles vary with topological charges of vortex beams, which inevitably inhibits the co-propagation of multiple OAM beams in communication systems. To overcome this issue, the concept of perfect vortex (PV) beams, whose annular intensity profiles is independent of topological charge, has been proposed [11,12]. Plenty of schemes to obtain PV beams have been proposed by utilizing a series of bulky optical elements, such as spatial light modulator, axicon, spiral phase plate, Fourier transform lens and so on. The above solutions require interaction of multiple optical devices, which hinders the miniaturization and integration of the OAM-based optical systems in practical application.
Metasurface, composed of nanostructures designed and arranged in specific patterns at the interface between two media, has attracted great research interest for its excellent capability of shaping the incident light by means of manipulating the phase profile through each nanostructure [13–20]. From the perspective of phase manipulation mechanism, both propagation-phase metasurfaces [21] and geometric-phase metasurfaces [22–24] have been proposed to generate PV beams with high efficiency or broad wavelength range. Geometric-phase metasurfaces, consisting of anisotropic antennas with identical shapes and different in-plane orientations, have advantages in fabrication tolerance and material property variations [25], and are usually applied to design compact flat components in area of holograms [26–29], metalenses [30–32] and so on. However, geometric-phase metasurface is always addressed with circularly or elliptically polarized light [33], which hinders the miniaturization of compact optical devices. Meanwhile, the symmetry of geometric-phase metasurface for two opposite spin states, for instance, a focusing metalens for left-handed circularly polarized (LCP) incidence is diverging for right-handed circularly polarized (RCP) incidence, may pose obstacles on the design of multifunctional integration device. The method of combining geometric phase with propagation phase is an approach in multifunctional device design, but it needs to scan a large number of parameters to obtain desired nanostructures.
Recently, Yuan et al. established chirality-assisted geometric-phase metasurfaces through introducing the concept of chirality-assisted phase as a degree of freedom. This type of phase modulation scheme activated all circularly polarized (CP) channels, and make full utilization of transmitted energy simultaneously, which could decouple the two co-polarized outputs, and thus be an alternative and potential solution for designing modulated-phase metasurfaces in multifunctional device design [34]. Zang et al. proposed an approach to realize multi-focus metalens based on geometric-phase metasurface when linear polarization light incident, and realized focusing and imaging in terahertz waveband [35]. As for vortex beam, it provides a new dimension for the modulation of the light due to the helical phase wavefront. Especially PV beam, as a special form of vortex beam it has pivotal applications in optical communication since its ring radius is independent of the topological charge. Therefore, it is of great significance to provide a more effective and flexible method to generate PV beam. In addition, so far there is no report on the generation of PV beam with polarization-rotated functionality, which is urgently needed in the field of information encryption because this feature will help to expand the dimension of the information carried by PV beam.
In this work, we designed a geometric-phase metasurface to generate PV beam for linearly-polarized (LP) light incident using the orthogonal decomposition of polarization vectors (ODPV). Without arranging multiple optical components or utilizing joint modulation (geometric phase and propagation phase), the designed metasurface is of single layer identical rectangular nanofins with different in-plane orientations, which endows the designed metasurface advantages of thin structure and convenience in manufacturing. Based on the principle of ODPV, we introduced the PV phase profiles corresponding to LCP component and RCP component into the metasurface, and determined the rotation angle of each nanostructure of the metasurface through calculating the argument of their composite vector in the transmission field. By setting the polarization rotation angle, PV beam with polarization-rotated functionality was demonstrated when LP incident. Furthermore, dual PV beams with orthogonal polarization states are obtained by means of superimposing two sets of phase profiles on a single metasurface at the same time. Besides, the ring radius and/or topological charge of PV beam can be set on demand. Although geometric-phase metasurface is spin-dependent, that is, typically working under circularly or elliptically polarized incident light [33], the proposed metasurface in our paper is working under LP incident. Breaking the spin-dependence, the proposed flexible and robust approach for generating PV beam with polarization rotation functionality has great potential for realizing optical path miniaturization in compact and integrated optical systems.
2. Theoretical analyses
2.1 Principle of PV beam
Vortex beam is a hollow beam with a helical phase of exp(imθ), where m is topological charge and θ is azimuth angle. PV beam is a kind of special vortex beam whose annular intensity profiles are independent of topological charge. In practice, PV beam can be generated through a metasurface possessing the superposition of the phase profiles of an axicon, a spiral phase plate and a Fourier transformation lens. Their corresponding phase profiles are defined by the following functions:
Here, x and y represent the coordinates of the nanostructure with respect to the center of metasurface. Equation (1) is the phase profile of axicon and d is the axicon period which relates to the ring radius of the PV beam; Eq. (2) shows the spiral phase profile of an optical vortex, where m is the topological charge; Eq. (3) gives the expression of a Fourier transformation lens, where λ is the designed working wavelength and f is the focal length of Fourier lens. Hence, the superimposed phase profile (as illustrated in Fig. 1) of the designed metasurface to generate PV beam can be expressed as:
Fig. 1. The phase profile of the designed metasurface which superimposes three-phase profiles of axicon, spiral phase plate and Fourier lens.
2.2 Principle of ODPV based geometric-phase metasurface
In this work, we adopt purely geometric-phase metasurface to generate PV beam. Geometric phase stems from the rotation of anisotropic nanostructure around the propagation direction relative to, say, x axis, and it has the form of ${\phi _{\textrm{PB}}} ={\pm} 2\sigma \theta$, where θ is rotation angle of nanostructure relative to x axis and σ is helicity of incident beam. Assuming that the long axis of the metasurface unit structure is paralleled to x-axis. When the x-polarized incidence passes through the unit structure array, the polarization of the transmitted beam will be the same as incident light. Taking advantage of orthogonal decomposition of polarization vectors (ODPV), that is, an LP beam can be decomposed into LCP component and RCP component with equal intensities and same initial phase, the Jones vector of the transmitted beam can be expressed as: $\left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1\\ \textrm{i} \end{array}} \right] + \frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1\\ { - \textrm{i}} \end{array}} \right]$, where ${\left[ {\begin{array}{*{20}{c}} 1&\textrm{i} \end{array}} \right]^\textrm{T}}$ represents the LCP component, and ${\left[ {\begin{array}{*{20}{c}} 1&{ - \textrm{i}} \end{array}} \right]^\textrm{T}}$ represents the RCP component. For geometric-phase metasurface, when the optical axis of a unit structure is rotated by angle $\theta (x,y)$ around the x-axis, the corresponding Jones matrix of the transmitted optical field is as follows:
Similarly, when the incidence is RCP beam, the transmitted light is described as:
Hence, upon the illumination of LCP beam or RCP beam, the transmitted field includes two parts, one is with same polarization (i.e. co-polarized) and the other is with orthogonal polarization (i.e. cross-polarized) as incident beam. If all unit structures are designed as local transparent half-wave plates, then the co-polarized part disappears, leaving cross-polarized part only. Under this thought, incidence of LP light can be converted into the superposition of cross-polarized parts in Eq. (6) and Eq. (7):
Therefore, the phase profile of the metasurface to generate PV beam with polarization rotate functionality can be described as:
Accordingly, the phase profile of the metasurface to generate dual PV beams with polarization rotate functionality can be described as [35]:
3. Simulation and discussions
To achieve the target phase profile in our simulations, the dielectric anisotropic TiO2 nanofin, which functions as a half-wave plate, is used for the construction of geometric-phase metasurface. As shown in Figs. 2(a-c), the rectangular nanofin sitting on the glass substrate is located at the centers of the unit cells with period P = 350 nm along both x-axis and y-axis. The optimized width W and length L of the rectangular nanofin is 82 nm and 274 nm, respectively, and the height H is 600 nm. The working wavelength is 532nm. And the rotation angle $\theta (x,y)$ of the rectangular nanofin in each unit cell is defined by $\theta (x,y) = {{\Phi (x,y)} / 2}$. We calculated the amplitude transmission coefficients (Tx and Ty) and the corresponding unwrapped transmitted phase (ϕx and ϕy) for the x- and y-polarized incident lights using FDTD solution, as shown in Fig. 2(d). The working wavelength (532 nm) is depicted with red vertical dashed line. At working wavelength, the proposed TiO2 unit cell exhibits phase retardation close to π (ϕx is −1.64 rad and ϕy is −4.82 rad), and the amplitude transmission coefficients Tx and Ty are all above 0.93, which ensured that the optimized nanofin has the function of a quasi-perfect half-wave plate.
Fig. 2. Schematic illustration of the designed unit cell. (a) Perspective view of the unit cell, the rectangular nanofin sitting on the glass substrate is located at the center of the unit cell, and the height H of the nanofin is 600 nm. (b) Top view of the unit cell, showing the rotation angle $\theta $ of the nanofin that leads to geometric-phase. (c)Top view of the unit cell, the width W and length L of the rectangular nanofin is 82 nm and 274 nm, respectively, and the period P is 350 nm. (d) Wavelength dependence of amplitude transmission coefficients (Tx and Ty) and the corresponding unwrapped transmitted phases (ϕx and ϕy) for the selected nanofin size under x- and y-polarized incidences. The working wavelength (532 nm) is depicted with red vertical dashed line, where the nanofin exhibits phase retardation close to π.
To demonstrate the characteristics of the generated PV beam, FDTD solution was used for all simulations, where perfectly matching layers (PMLs) were used in x, y, and z directions, and x-polarized light beam was used as the incident light. All the designed metasurfaces are consisting of 80 nanofins along both x and y directions. And the method of far-field extraction was used to calculate the electric field intensity of the ring after the incident light passes through the metasurface under normal incidence.
3.1 Generation of single PV beam
In this section, firstly, we designed two sets of metasurfaces to generate PV beam without polarization rotation [the polarization rotation angle $\phi = 0$ in Eq. (10)]. The theoretical focal length f = 100µm for different topological charges (m = 2, 3, 5, 7). The axicon periods d = 4µm and d = 6µm, respectively. Figure 3 shows the top views and the details of the designed metasurfaces for topological charge m = 2, where the axicon periods d = 4µm in 3(a) and d = 6µm in 3(b).
Fig. 3. The top views and the details of the designed metasurfaces for topological charge m = 2, where the axicon periods (a) d = 4 µm and (b) d = 6 µm.
Figure 4(a) presents the simulated intensity profiles of the generated PV beams at distinct propagation distances. It is shown that the transmitted beam gradually converges into an annular ring at the position of z = 70µm. As the PV beam further propagates in the free space, the radius of annular ring becomes larger while the circular shape is being kept. Figures 4(b) and 4(d) show the intensity distributions of PV beams at propagation distance of z = 100µm for different topological charges, where the axicon period is 4µm in Fig. 4(b) and 6µm in Fig. 4(d). In our simulations, since all the designed metasurfaces are in square-shape, i.e., consisting of 80 nanofins along both x and y directions, the interference of the light fields provided by the four corners of the metasurface reduce the uniform of the intensity distributions of PV beams. Using circular-shape arrangement or increase the size of the metasurface will improve the uniform of the bright ring. Considering the lower polarization rotation efficiency of circular arrangement and the limitation of the computer, we adopted metasurface in square-shape arrangement with $80 \times 80$ nanofins for all simulations. The extracted corresponding cross-sections of the normalized intensity profiles along the x-direction are shown in Figs. 4(c) and 4(e), respectively. Here, we use the distance of the maxima of the extracted intensity profiles in cross-sections to calculate the radius of the ring, as shown in Figs. 4(c) and 4(e), where D is the diameter of the ring. In Fig. 4(c), the radii of the rings are 14µm, 14.8µm, 15.3µm, 15.5µm, and in Fig. 4(e), the radii of the rings are 10µm, 11µm, 11µm, 11.8µm, respectively for topological charges m = 2, 3, 5, 7. The reason for the increment of the ring radius is that the number of nanofins affects metasurface’s constrain ability over the radius of the bright ring. More nanofins of the metasurface can effectively alleviate the variation of ring radius. We can see that under a fixed axicon period, although the radii of the rings have slight increase, it still exhibits the characteristic that the radius of the PV beam is independent of the topological charge, indicating that PV beam can be generated by geometric-phase metasurface in the case of LP incident. This breaks the rule that geometric-phase metasurface must be addressed with circularly or elliptically polarized light [33] and realizes spin-decoupled. The proposed method may provide promising opportunities for designing polarization-dependent planar optical devices and facilitating optical path miniaturization in compact and integrated optical systems.
Fig. 4. (a) The intensity distributions of the generated PV beam of axicon period d = 4 µm and topologic charge m = 2 at distinct propagation distances. The transmitted beam gradually converges into an annular ring at the position of z = 70 µm. As the PV beam further propagates along z-axis, the radius of annular ring becomes larger while the circular shape is being kept. (b, d) The intensity distributions of generated PV beams at propagation distance of z = 100 µm for different topological charges, where the axicon periods are d = 4 µm and 6 µm, respectively. (c, e) The extracted corresponding cross-sections of the normalized intensity profiles along x-direction, where D is the diameter of the ring. It is observed that under a fixed axicon period, the ring radius of the generated PV beams almost remain unchanged as the change of topological charges. As the axicon period increases, the radius of the ring decreases.
The relationship between the ring radius of the PV beam and propagation distance can help us predict the size of the bright ring at different locations more accurately, which is very useful in designing PV beam-based metasurfaces. Figure 5 shows the simulated ring radius r of the generated PV beam as a function of the propagation distance for metasurfaces with m = 2, d = 4 and 6µm, respectively. The simulation results show that the ring radius is almost proportional to propagation distance z, and the fitting function can be expressed as:
where the lens focal length is f = 100µm, the unit of all variables is µm. In fact, in addition to the parameters in the fitting function Eq. (12), the ring radius is also related to the incident wavelength, which is fixed in our all simulations.Fig. 5. Simulated ring radius of generated PV beam versus the propagation distance z (star point), and the fitted lines for simulated data (solid lines). The parameters of metasurface are: (a) the axicon period d = 4 µm and topologic charge m = 2; (b) the axicon period d = 6 µm and topological charge m = 2.
Fig. 6. The top views and the details of the designed metasurfaces to generate single PV beam with polarization rotation for different topological charges (m = 2, 3, 5, 7), where the axicon periods d = 4 µm.
In the following we verify that by setting the polarization rotation angle, PV beam with polarization rotation functionality is realized. Here we designed another set of geometric-phase metasurface with the polarization rotation angle $\phi = 90$ in Eq. (10). The structure parameters are as follows: theoretical focal length of Fourier lens f = 100µm, axicon period d = 4µm and topologic charges m = 2,3,5,7, respectively, and Fig. 6 display the top views and the details of the designed metasurfaces. Figure 7 display the intensity distributions in x-y plane (${|{{{\textbf E}_y}} |^2}$ and ${|{{{\textbf E}_x}} |^2}$) of PV beams at propagation distance of z = 100µm for different topological charges. Notice that the polarization rotation angle is 90 degree, the converted y-polarized electric field contributes to the bright annular ring for x-polarized incidence. We calculated the radii of the PV beams in Fig. 7(a), and the results are 14µm, 14.5µm, 15.2µm, 16.2µm, for m = 2, 3, 5, 7, respectively. Although the radii of the rings have slightly increased with the increase of topological charge, it still exhibits the characteristic of the PV beam which the radius of the ring is independent of the topological charge. In our simulations, all the polarization conversion efficiencies (defined as the ratio of the intensity of converted polarized component to the intensity of total transmitted beam) were larger than 72.3%. The loss of energy may come from several aspects, for example, the phase difference between the selected nanofin and the half-wave plate, which is 2.2 degree in our simulations. What’s more, due to the memory limitation of the far-field extraction method, only 41 pixels are involved in the calculation of electric field intensity along both x direction and y direction. Nevertheless, the majority of the transmitted field is the converted electric field component, i.e. y-polarized electric field, which contributes to the bright annular ring. More importantly, the ring radius almost not varies with the change of topological charge, which indicates that the PV beam with polarization rotation functionality can be realized by geometric-phase metasurface in the case of LP incident.
Fig. 7. (a) The intensity distributions of ${|{{{\textbf E}_y}} |^2}$ in x-y plane of generated PV beam at propagation distance of z = 100µm for different topological charges. Because the polarization rotation angle is 90 degree, the converted y-polarized electric field contributes to the bright annular ring when x-polarized incidence. (b) The intensity distributions of ${|{{{\textbf E}_x}} |^2}$ in x-y plane of generated PV beam at propagation distance of z = 100µm for different topological charges.
3.2 Generation of dual PV beams
The above simulations verify the generation of PV beam by geometric-phase metasurface when LP incident. In this section, we prove that dual distinct PV beams can be achieved by one geometric-phase metasurface and some features of PV beam can be set flexibly on demand, which may open an avenue for designing multi-functional metasurfaces and shared-aperture devices.
According to Eq. (11), we design a metasurface of theoretical focal length f = 100µm and topological charge m = 2. Axicon period d = 4µm and polarization rotation angle ${\phi _1} = 0$ are adopted for the first PV beam phase ${\varphi _1}(x,y)$; axicon period d = 6µm and polarization rotation angle ${\phi _2} = 90$ are adopted for the second PV beam phase ${\varphi _2}(x,y)$. Figure 8(a) shows the top views and the details of the designed metasurface.
Fig. 8. The top views and the details of the designed metasurfaces to generate dual distinct PV beams with orthogonal polarization states.
According to theoretical analysis, dual PV beams with distinct ring radii and different polarizations (polarization along x-axis for the first PV beam and y-axis for the second PV beam) can be achieved simultaneously when x-polarized light incident. Figures 9(a) and 9(b) show the intensity profiles of ${|{{{\textbf E}_x}} |^2}$ and ${|{{{\textbf E}_y}} |^2}$ in the transmitted field at propagation distance of z = 100µm respectively. It can be seen that almost all the converted electric field component, i.e. y-polarized electric field contributes to the inner ring (the second PV beam), while the unconverted electric field component, i.e. x-polarized electric field contributes to the outer ring (the first PV beam). The proportions of x-polarized electric field component and y-polarized electric field component in the total transmission field are 50.77% and 47.14%, respectively. Figure 9(c) gives the corresponding cross-sections of the normalized intensity profiles extracted along the x-direction, indicating that the ring radius is related to axicon period and decreases as the axicon period increases. In the meantime, since the outer ring and the inner ring have almost equal proportions, and the radius of the outer ring is relatively large, the peak intensity is smaller than that of the inner ring while the normalized intensity of both PV beams are greater than 0.7. In general, all above simulated results agree well with theoretical prediction, confirming that the proposed metasurface has the function of generating dual PV beams with orthogonal polarization states.
Fig. 9. (a) The intensity distributions ${|{{{\textbf E}_x}} |^2}$ of generated first PV beam at propagation distance of z = 100µm for topological charges m = 2 and axicon period d = 4 µm. (b) The intensity distributions ${|{{{\textbf E}_y}} |^2}$ of generated second PV beam at propagation distance of z = 100µm for topological charges m = 2 and axicon period d = 6 µm. (c) The corresponding cross-sections of the normalized intensity profiles extracted along x-direction.
In the design of metasurface, flexible parameter settings are always desirable. In the following we verify that some features of the generated PV beam, such as ring radius, topological charges, can be set flexibly to meet different application requirements. We designed two metasurfaces according to Eq. (11). Meanwhile, to show the function of polarization rotation, we set ${\phi _1} = 0$ and ${\phi _2} = 90$.The first metasurface is designed to generate two PV beams with same ring radius but different in topological charge, as shown in Fig. 10(a) [The top view and details are shown in Fig. 8(b)]. This kind of PV beam could be applied in optical tweezers [36]. As we know, vortex beam has gradual phase gradient, which can generate phase gradient force to rotate the particles. The larger the topological charge carried by the vortex beam, the larger the relative gradient force and the faster the speed of particle moving. In addition to setting the topological charges of the two generated PV beams only, the ring radius and topological charge can also be set different at the same time. To this end, we design another metasurface with different ring radius and topological charge, as shown in Fig. 10(b) [The top view and details are shown in Fig. 8(c)]. This function shows advantage in the application of optical particle trapping [37], because different ring radii have different intensity gradient forces, which can trap particles near the center of the focusing spot. Therefore, the two vortex beams with different radii and different topological charges can control the particles in the optical field to make them in different states simultaneously. In our simulations, all the transmission efficiencies (defined as the ratio of the power in the transmitted field to the total incident beam power) were larger than 0.36, and transmission efficiencies is slightly different between matesurfaces with and without polarization rotation functionality.
Fig. 10. (a) The intensity distributions of generated two PV beams at z = 100 µm for same ring radius but different topological charges. (b) The intensity distributions of generated two PV beams at z = 100 µm for different ring radius and different topological charges.
4. Conclusion
In summary, we proposed a geometric-phase metasurface to generate PV beam upon the illumination of LP light. The method of generating PV beam breaks the spin-dependence of geometric-phase metasurface, therefore can reduce optical path in compact miniature optical systems effectively. Based on the principle of ODPV, one PV beam with polarization-rotated functionality is obtained by imposing phase profiles corresponding to LCP component and RCP component to the designed metasurface. Furthermore, dual PV beams with orthogonal polarization states are realized when two sets of phase profiles patterned on a single metasurface simultaneously. Benefiting from the unique property of the proposed metasurface, the ring radius and/or topological charge of PV beam can be set on demand flexibly. This novel type of PV beam generator will find potential applications in compact and integrated optical systems.
Funding
National Natural Science Foundation of China (61975125); Leading Academic Discipline Project of Shanghai Municipal Government (S30502).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the authors on reasonable request, see author contributions for specific data sets.
References
1. K. Zhang, Y. Wang, S. N. Burokur, and Q. Wu, “Generating dual-polarized vortex beam by detour phase: from phase gradient metasurfaces to metagratings,” IEEE Trans. Microwave Theory Techn. 70(1), 200–209 (2022). [CrossRef]
2. M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013). [CrossRef]
3. S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102(11), 113602 (2009). [CrossRef]
4. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef]
5. F. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
6. D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013). [CrossRef]
7. 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. Photonics 7(1), 66–106 (2015). [CrossRef]
8. B. C. Hiesmayr, M. J. A. De Dood, and W. Löffler, “Observation of four-photon orbital angular momentum entanglement,” Phys. Rev. Lett. 116(7), 073601 (2016). [CrossRef]
9. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
10. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012). [CrossRef]
11. A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizon, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38(4), 534–536 (2013). [CrossRef]
12. P. Vaity and L. Rusch, “Perfect vortex beam: fourier transformation of a bessel beam,” Opt. Lett. 40(4), 597–600 (2015). [CrossRef]
13. A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015). [CrossRef]
14. M. Tseng, A. Jahani, A. Leitis, and H. Altug, “Dielectric metasurfaces enabling advanced Optical biosensors,” ACS Photonics 8(1), 47–60 (2021). [CrossRef]
15. H. Zhang, Y. Liu, Z. Liu, X. Liu, G. Liu, G. Fu, J. Wang, and Y. Shen, “Multi-functional polarization conversion manipulation via graphene-based metasurface reflectors,” Opt. Express 29(1), 70–81 (2021). [CrossRef]
16. W. Yang, S. Xiao, Q. Song, Y. Liu, Y. Wu, S. Wang, J. Yu, J. Han, and D. Tsai, “All-dielectric metasurface for high-performance structural color,” Nat. Commun. 11(1), 1864 (2020). [CrossRef]
17. Z. Yue, J. Liu, J. Li, J. Li, C. Zheng, G. Wang, M. Chen, H. Xu, Q. Wang, X. Xing, Y. Zhang, Y. Zhang, and J. Yao, “Multifunctional terahertz metasurfaces for polarization transformation and wavefront manipulation,” Nanoscale 13(34), 14490–14496 (2021). [CrossRef]
18. S. Tian, H. Guo, J. Hu, and S. Zhuang, “Dielectric longitudinal bifocal metalens with adjustable intensity and high focusing efficiency,” Opt. Express 27(2), 680–688 (2019). [CrossRef]
19. J. Hu, J. Xie, S. Tian, H. Guo, and S. Zhuang, “Snell-like and Fresnel-like formulas of the dual-phase-gradient metasurface,” Opt. Lett. 45(8), 2251–2254 (2020). [CrossRef]
20. Z. Qian, S. Tian, W. Zhou, J. Wang, and H. Guo, “Broadband achromatic longitudinal bifocal metalens in the visible range based on a single nanofin unit cell,” Opt. Express 30(7), 11203–11216 (2022). [CrossRef]
21. J. Xie, H. Guo, S. Zhuang, and J. Hu, “Polarization-controllable perfect vortex beam by a dielectric metasurface,” Opt. Express 29(3), 3081–3089 (2021). [CrossRef]
22. Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018). [CrossRef]
23. Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017). [CrossRef]
24. Q. Zhou, M. Liu, W. Zhu, L. Chen, Y. Ren, H. J. Lezec, Y. Lu, A. Agrawal, and T. Xu, “Generation of perfect vortex beams by dielectric geometric metasurface for visible light,” Laser Photonics Rev. 15(12), 2100390 (2021). [CrossRef]
25. B. Yao, X. Zang, Y. Zhu, D. Yu, J. Xie, L. Chen, S. Han, Y. Zhu, and S. Zhuang, “Spin-decoupled metalens with intensity-tunable multiple focal points,” Photonics Res. 9(6), 1019–1032 (2021). [CrossRef]
26. Z. Yue, G. Xue, J. Liu, Y. Wang, and M. Gu, “Nanometric holograms based on a topological insulator material,” Nat. Commun. 8(1), 15354 (2017). [CrossRef]
27. L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015). [CrossRef]
28. L. Jin, Z. Dong, S. Mei, Y. Yu, Z. Wei, Z. Pan, S. Rezaei, X. Li, A. I. Kuznetsov, Y. S. Kivshar, J. K. W. Yang, and C. W. Qiu, “Noninterleaved metasurface for (26-1) spin- and wavelength-encoded holograms,” Nano Lett. 18(12), 8016–8024 (2018). [CrossRef]
29. Y. Wang, C. Guan, H. Li, X. Ding, K. Zhang, J. Wang, S. N. Burokur, J. Liu, and Q. Wu, “Dual-polarized tri-channel encrypted holography based on geometric phase metasurface,” Adv. Photon. Res. 1(2), 2000022 (2020). [CrossRef]
30. M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352(6290), 1190–1194 (2016). [CrossRef]
31. W. Chen, A. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13(3), 220–226 (2018). [CrossRef]
32. R. Lin, V. Su, S. Wang, M. Chen, T. Chung, Y. Chen, H. Kuo, J. Chen, J. Chen, Y. Huang, J. Wang, C. Chu, P. Wu, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “Achromatic metalens array for full-colour light-field imaging,” Nat. Nanotechnol. 14(3), 227–231 (2019). [CrossRef]
33. P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017). [CrossRef]
34. Y. Yuan, K. Zhang, B. Ratni, Q. Song, X. Ding, Q. Wu, S. N. Burokur, and P. Genevet, “Independent phase modulation for quadruplex polarization channels enabled by chirality-assisted geometric-phase metasurfaces,” Nat. Commun. 11(1), 4186 (2020). [CrossRef]
35. X. Zang, H. Ding, Y. Intaravanne, L. Chen, Y. Peng, J. Xie, Q. Ke, A. V. Balakin, A. P. Shkurinov, X. Chen, Y. Zhu, and S. Zhuang, “A multi-foci metalens with polarization-rotated focal points,” Laser Photonics Rev. 13(12), 1900182 (2019). [CrossRef]
36. W. Wang, C. Jiang, H. Dong, X. Lu, J. Li, R Xu, Y. Sun, L. Yu, Z. Guo, X. Liang, Y. Leng, R. Li, and Z. Xu, “Hollow plasma acceleration driven by a relativistic reflected hollow laser,” Phys. Rev. Lett. 125(3), 034801 (2020). [CrossRef]
37. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef]
