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Abstract
Orbital angular momentum (OAM) beams have recently attracted attention due to their intrinsic properties and their many novelty applications. The use of dielectric metasurfaces as highly efficient and ultra-thin functional optical platforms for OAM beam generation has been discussed in detail in earlier research. However, reflective all-dielectric metasurfaces for high quality OAM beam generation have been rarely investigated. In this work, a high-quality OAM beam generator applying an all-dielectric reflective metasurface is proposed, with an operation waveband from 1500 to 1600 nm. Simulation results show that good quality OAM beams (mode purity ∼99%) can be generated with high efficiency (conversion rate >97%) and high reflectance (>0.8). With such superior performance, the proposed reflective all-dielectric metasurface can provide new possibilities in free-space optical communication systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortex beams carrying an orbital angular momentum (OAM), which are also referred as OAM beams [1], have attracted great attention due to its fascinating properties, and the discovery of vortex beams also leads to an improvement of optical systems. In addition, OAM beams have been broadened to numerous applications, such as particle trapping [2,3], optical communications [4–6], beam focusing [7] and so on. Meanwhile, the methods for generating OAM beams also become the focusing point and have been extensively investigated. A variety of OAM beam generation devices, including spiral phase plate (SPP) [8,9], spatial light modulator (SLM) [10], computer generated hologram [11] and metamaterial/metasurface [12–20], have been numerically or experimentally reported. However, conventional methods, like SPP, SLM, hologram or metamaterial are always bulky or fragile, and thus hard to integrate into nanoscale optoelectronic devices.
Metasurface as a reliable alternative to conventional devices, which is composed of single subwavelength structure, not only retains efficient control for phase, polarization and amplitude of light, but also enables highly compact optical devices with different functionalities [21,22]. With metasurfaces, a wide range of devices supporting phase and polarization manipulation have also been realized such as wave plates [23–25], lens [26–28] and gratings [29]. The OAM beam generators working in both transmissive and reflective modes have also been investigated through different designs [12–20]. However, among them most of reflective devices are composed of metallic structures or applying a metallic substrate/ground. The utilization of plasmonic structures can cause severe losses and greatly reduce the device efficiency, especially in visible and near-infrared region [30].
Dielectric metasurfaces hold promise for various applications because of their ultra-compact, high-efficiency and reduced fabrication complexity [31,32]. Complete control of polarization and phases based on all-dielectric metasurface has already been proposed [33]. Moreover, dielectric metasurfaces for OAM beam generation have also been realized and covers different wave regions, including the optical regime [12,13] and the microwave region [14–20]. However, these designs are either work in transmission mode or hard to implement high quality beam generation with high efficiency. Mode purity as one of the crucial properties of OAM beams, is of great significance in free-space optical communication systems [34] and many other applications [35,36]. For OAM beams generated by metasurface, the dense arrangement of resonators and multi-layer structure may lead to the variations of the refractive index along the transmission path which may distort the helical wave front and increase the bit error rate (BER) of data transmission in optical communication systems, as turbulence usually does [37]. The lower mode purity is, the higher cross-talk and BER will be. Hence, it is significant to generate OAM beams by metasurface with high mode purity.
In this paper, we numerically demonstrate an all-dielectric metasurface for high quality and high-efficiency OAM beam generation. The conceptual illustrations of the proposed metasurface are shown in Fig. 1. With circular polarized light incident, the metasurface is capable to reflect beam carrying an OAM. For left and right circular light (LCP and RCP) incident, the output OAM states are constrained to hold equal or opposite topological charges l and –l. Here, we propose a novel approach to generate OAM beams using all-dielectric reflective metasurface, while maintaining high beam quality and high-power efficiency. Besides efficient beam generation, wide operation bandwidth is also significant for practical applications. Simulation results show that generated OAM beam with l = –1 has mode purity above 99% in the telecom waveband from 1500-1600 nm, and for l = –20, mode purity is still as high as 97%. The lowest reflectance for the metasurface is above 0.81 in the wave ranges from 1500-1600 nm, and highest reflectance can reach 0.94. With low losses and reflective design, the proposed metasurface could pave new avenues for light manipulation in large-area photonic applications.
Fig. 1. Conceptual schematic of the all-dielectric reflective metasurface for OAM beam generation. The metasurface operates on a circular polarization basis (|L〉 and |-R〉) and the generated OAM beam can carry arbitrary topological charge l, depending on resonator arrangements.
2. Structures and physical mechanism
A schematic illustration of the reflective all-dielectric metasurface is shown in Fig. 2. It is composed of silicon elliptical nanopillars and a fused silica substrate. The basic structure of a resonator with detailed geometric definition is plotted in Fig. 2(b). The reflective all-dielectric metasurface is based on Mie resonance principle but different from transmissive metasurafce. By utilizing elliptical nanopillar with an aspect ratio of approximately 1:1 while keeping electric and magnetic resonances excitation, high reflectance can be obtained in the desired waveband. In our work, geometric parameters of the resonator are optimized to meet the conditions mentioned above and realize high reflectance as well as high polarization conversion efficiency. We employ finite-difference time-domain (FDTD) method provided by Lumercial to give precise results. In simulations, the background has a refractive index constant as 1, silicon and fused silica have wavelength dependent refractive index adopted from optical data of Palik (1.0-2.0 µm).
Fig. 2. (a) Schematic of the proposed all-dielectric reflective metasurface. (b) Geometric illustration of a unit cell, with periodicity Px = Py = 800 nm, resonator diameter DL = 720 nm, DS = 320 nm, height H = 560 nm and orientation angle θ = 45°. (c) Schematic of polarization conversion. EL and ES are the decomposition components of the incident and reflected linear polarized light along long and short axes of the ellipse resonator respectively.
To realize the reflective OAM beam generator we propose, the reflective metasurface should apply an azimuthally dependent phase pattern, and thus a reflected beam is able to have a helical wave front and phase singularity, which can be regarded as a typical OAM beam. Hence, each resonator needs to realize a gradient phase control from 0 to 2π, allowing us to design customized phase pattern. Geometric phase manipulation, which is different from adjusting cut-wire geometry of the resonator to implement phase control, is a robust method to imprint phase to the circular polarized (CP) light by simply rotating the orientation of the resonator. On the basis of geometric control principle [38], the resonator is desired to operate as a half wave plate (HWP), to maximize manipulate efficiency. In our proposed reflective metasurface, by rotating the long axis of the resonator with angle θ, when a CP light incident, the reflect beam will be imprinted with a geometric phase equal to φ = 2·θ and maintains the same helicity.
Thus, to realize polarization conversion and impart a π phase shift to incident wave, the electric and magnetic dipole resonances need to be optimized by finely tuning geometric parameters of the basic resonator. The optimized geometric parameters are DL = 720 nm, DS = 320 nm, H = 560 nm with orientation angle θ = 45°, and periodicities along x and y axis are Px = Py = 800 nm. And to further characterize the basic functionalities of the all-dielectric reflective array, we have x-pol incident on the structure and test how well the resonator can convert incident light into an orthogonal polarization. We use polarization conversion rate (PCR) to evaluate conversion efficiency, which is defined as:
where Rcross and Rco are the cross-and co-polarization reflectance. The simulation results show that the PCR remains above 80% in a broad waveband from 1300 to 1700nm, as shown in Fig. 3(b) (except for a grey shaded region, near 1440 nm). And in the working bandwidth from 1500 to 1600 nm, the PCR is above 97%. To better illustrate the response of the resonator, we decompose the incident x-pol light and reflected y-pol light into two orthogonal components EL and ES as depicted in Fig. 3(c), and the directions of EL and ES components are along the long and short axes of the resonator, respectively. With x-pol incident, the reflected EL component is assumed to maintain its intensity and direction, while ES component is reversed, due to the π phase difference. Simulated reflectance and phase spectra for a resonator with polarizations incident along long and short axis are plotted in Fig. 3(c)-(d). The numerical simulations demonstrate that the reflectance for both polarizations are near-unity while maintaining a π phase difference throughout a broad waveband from 1500 to 1600 nm, leading to an orthogonal polarization conversion with high efficiency when light incident along x- or y-axis.Fig. 3. (a) Simulated reflectance for co- and cross-polarized light with x-pol incident. (b) Polarization conversion rate of the reflective array. (c) Reflectance and (d) phases for EL (blue curve) and ES (red curve) polarized light incident, demonstrating near-unity reflectance in several spectral range and a broadband π phase difference.
However, it should be notice that the reflection dips to 0 near 1440 nm when light incident along short axis of elliptical resonator, which further narrow the working bandwidth of the metasurface. To confirm the cause of the reflection dip and identify the involved resonance modes, we employed the framework of Mie theory [39,40] for mode calculation, and gave the near-field distributions to corroborate the excitation of these modes. Calculated reflection spectra for linear polarized (LP) incident along long and short axis, with each peak corresponding to a specific resonance type, are plotted in Fig. 4(a) and (b), respectively. In Fig. 4(a), it shows that for LP incident along long axis, the peaks near 1410 nm and 1590 nm are corresponding to a magnetic dipole (MD) resonance mode and an electric dipole (ED) resonance mode, which are the dominant contribution to backward scattering efficiency. As a comparison, Fig. 4(b) shows that, for LP incident along short axis, a magnetic quadrupole resonance is excited near 1380 nm, and an ED resonance is excited at a longer wavelength near 1530 nm, while the two peaks are narrower and further separated. Since the reflection mainly results from backward scattering that contributed by these resonance modes, the scarcely overlapping resonance modes is one of the reasons for the reflection dip, which further affects the working bandwidth. On the other hand, when electric and magnetic dipole coefficients coincident, the backward scattering nearly vanishes and leads to negligible reflection. This situation happens when both the wavelength of incident light and the size parameter of scattering particle satisfy the conditions for a minimal ratio of the backward and forward scattering, often called the first Kerker condition [41].
Fig. 4. Reflection spectrum and the multipole scattering efficiencies for LP incident along (a) long and (b) short axis.
To clearly clarify these resonance modes, the near-field distributions at 1440 nm are also plotted, as shown in Fig. 5(a)-(d). Here, to better understand the formation of electric and magnetic resonance, the resonator can be seen as a waveguide with different effective refractive indices along the long and short axis. When LP incident along two ellipse diameters (long and short axis), displacement currents are excited and further form different near-filed distribution inside the resonator, which can be determined as different resonance modes. In Fig. 5(a), with LP incident along long axis, the excited circular displacement currents form a maximum magnetic field at the center of elliptical nanopillar, oriented in the direction of the incident magnetic field. This is the typical feature of a MD mode. And in Fig. 5(c), the electric field distribution in the xy-plane is plotted, two strong electric field are formed by displacement currents at both sides of the elliptical cross-section, which can also be determined as a MD response. In contrast, with LP incident along short axis, the magnetic resonance is barely excited, as shown in Fig. 5(b). And in Fig. 5(d), the electric field distribution shows the feature of an ED resonance, with displacement current vectors inside the resonator pointing in the same direction to the incident electric field. Although the ED resonance is excited, it is too weak to contribute to the backward scattering. As shown above, when light incident along short axis, due to the lack of resonance contributing backward scattering, the reflection dips near 1440 nm, which affects the conversion efficiency and narrow the working bandwidth.
Fig. 5. The magnetic field distribution and displacement currents vectors (white arrows) in xz-plane at 1440 nm with LP incident along long and short axis, (a) and (b). The electric field distribution and displacement currents vectors (white arrows) in xy-plane at 1440 nm with LP incident along long and short axis, (c) and (d). Dash line depicts the boundaries of the resonator.
To realize the dielectric metasurface for OAM beam generation we propose, it is vital to validate the geometric phase control capability of the optimized reflective array. The phase and reflectance spectra at different resonator rotation angles of the reflected beam, that cover the telecom waveband from 1500 to 1600 nm, are plotted in Fig. 6. Within this bandwidth, the reflectance remains above 0.85 showing that the energy efficiency is nearly unaffected at various wavelengths as well as rotation angles. In addition, we can see that a smooth reflected phase increases from –π to π can be obtained with rotation angle increasing from 0 to π linearly. Therefore, the reflective metasurface has met the request to design required phase pattern and can be used for OAM beam generation.
Fig. 6. Simulated beam (a) geometric phase and (b) reflectance for the resonator with varying rotation angles at a fixed waveband from 1500 to 1600 nm.
3. Results and discussion
Based on the proposed reflective array and geometric phase control principle, we are able to generate OAM beams in a broad telecom wave band. The phase pattern for an OAM beam, featured with a helical wave front and carrying helical phase term ejlφ, can be expressed as:
where l is topological charge; x and y are coordinates in cartesian coordinates. When the rotation angles of resonators are arranged properly following Eq. (2), an incident RCP beam is transformed into vortex beam with corresponding l. The simulated results at the wavelengths of 1500, 1550 and 1600 nm with vortex beams l = –1, generated by the reflective metasurface, are shown in Fig. 7. As shown in Fig. 7(a)-(f), the phase distributions of the reflected wave front are consistent with designed phase pattern, and the electric field intensities have a doughnut shape, due to the phase singular point along propagation axis. By employing a mode decomposition method suitable for arbitrary light fields [42], the mode spectra of the vortex beam with l = –1 are also calculated for all wavelengths, as shown in Fig. 7(g)-(i). To further quantify the beam purity, we define mode purity as: where Im is the light intensity of the m-th mode. The quantified mode purity for all the three wavelengths and total reflectance of the reflective metasurface are summarized in Table 1. As we can see, the obtained mode purity is higher than 99.9% and reflectance is higher than 0.9, ensuring the highly efficient of the all-dielectric metasurface.Fig. 7. Normalized electric field intensity, phase distribution and mode spectra of the vortex beams with topological charge l = –1 at different wavelengths: (a), (d) and (g) for 1500 nm; (b), (e) and (h) for 1550 nm; (c), (f) and (i) for 1600 nm.
Table 1. Calculated Total Reflectance of the Metasurface and Mode Purity of the Vortex Beams with l = –1.
Broadband, high order vortex beam generation have been rarely investigated and are always hard to guarantee beam quality. However, in our case, OAM beams with high orders can also be generated by the all-dielectric reflective metasurface, when we follow the corresponding phase pattern as written in Eq. (2) to arrange resonators. In consequence, we simulated the reflective metasurface with l = –20 at different incident wavelengths, and the corresponding electric field as well as phase distribution are shown in Fig. 8(a)-(c) and (d)-(f), respectively. To better evaluate the mode purity, the calculated mode spectra were expanded to ± 30 orders, as shown in Fig. 8(g)-(i). As a further description for the performance of the metasurface, the total reflectance and mode purity of different wavelengths are illustrated in Table 2. However, owing to the complex phase distribution at the center of the metasurface and the mutual coupling between each resonator, intensity and mode purity for higher order OAM beam are lower than OAM beam with l = –1. With shorter periodicities for a unit cell and applying more units in the metasurface within the same plate size, the reflectance may be further increased while maintains high beam purity [43].
Fig. 8. Normalized electric field intensity, phase distribution and mode spectra of the vortex beam with topological charge l = –20 at different wavelengths: (a), (d) and (g) for 1500 nm; (b), (c) and (h) for 1550 nm; (c), (f) and (i) for 1600 nm.
Table 2. Calculated Total Reflectance of the Metasurface and Mode Purity of the Vortex Beams with l = –20.
In Table 3, we list detailed comparison of this work with previous similar works on reflective OAM generators in terms of some pivotal factors. Although the metasurface working in telecom waveband from 1500 to 1600 nm is not the first time to be investigated, this work exhibits higher power efficiency and mode purity than previous works. Moreover, when generating higher order OAM beams, the power efficiency and mode purity of the generator stay at a high level. Due to the dip appearing in the reflection and conversion spectra, the bandwidth is narrower than some previous work. However, it is possible to extend the bandwidth and fix the dip of the current work by applying other meta-atoms in a unit cell [28].
Table 3. Comparison with Previous Works.
4. Conclusions
In conclusion, we have numerically demonstrated an all-dielectric reflective metasurface for high quality OAM beam generation (mode purity ∼99%) that covers an optical telecom spectral range from 1500 to 1600 nm. The generator is composed of subwavelength dielectric resonators. Based on our novel approach, OAM beams carrying arbitrary topological charge can be generated depending on specific resonator arrangement in the wave range. Meanwhile, the proposed resonator unit can also work as a reflective half-wave plate, near-unity reflectance and high conversion efficiency can also be achieved in the same telecom waveband, that guarantee numerous applications in optoelectronic fields. As mentioned above, we believe this low-loss and highly efficient design will make a step forward in the development of advanced optoelectronic devices and free-space optical communications systems.
Funding
National Key Research and Development Program of China (2017YFB0403602); National Major Scientific Instruments and Equipment Development Project supported by National Natural Science Foundation of China (61827814).
References
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(11), 8185–8189 (1992). [CrossRef]
2. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010). [CrossRef]
3. B. Tian and J. Pu, “Tight focusing of a double-ring-shaped, azimuthally polarized beam,” Opt. Lett. 36(11), 2014–2016 (2011). [CrossRef]
4. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef]
5. Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40(21), 4843–4846 (2015). [CrossRef]
6. X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012). [CrossRef]
7. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef]
8. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996). [CrossRef]
9. P. Schemmel, G. Pisano, and B. Maffei, “Modular spiral phase plate design for orbital angular momentum generation at millimetre wavelengths,” Opt. Express 22(12), 14712–14726 (2014). [CrossRef]
10. A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38(4), 534–536 (2013). [CrossRef]
11. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]
12. M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015). [CrossRef]
13. Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric Meta-Reflectarray for Broadband Linear Polarization Conversion and Optical Vortex Generation,” Nano Lett. 14(3), 1394–1399 (2014). [CrossRef]
14. F. Bi, Z. Ba, and X. Wang, “Metasurface-based broadband orbital angular momentum generator in millimeter wave region,” Opt. Express 26(20), 25693–25705 (2018). [CrossRef]
15. R. Niemiec, C. Brousseau, K. Mahdjoubi, O. Emile, and A. Ménard, “Characterization of an OAM flat-plate antenna in the millimeter frequency band,” IEEE Antennas Wirel. Propag. Lett. 13, 1011–1014 (2014). [CrossRef]
16. H. Xu, H. Liu, X. Ling, Y. Sun, and F. Yuan, “Broadband vortex beam generation using multimode Pancharatnam–Berry metasurface,” IEEE Trans. Antennas Propag. 65(12), 7378–7382 (2017). [CrossRef]
17. C. Ji, J. Song, C. Huang, X. Wu, and X. Luo, “Dual-band vortex beam generation with different OAM modes using single-layer metasurface,” Opt. Express 27(1), 34–44 (2019). [CrossRef]
18. Z. Ma, S. M. Hanham, P. Albella, B. Ng, H. T. Lu, Y. Gong, S. A. Maier, and M. Hong, “Terahertz All-Dielectric Magnetic Mirror Metasurfaces,” ACS Photonics 3(6), 1010–1018 (2016). [CrossRef]
19. S. Kruk, B. Hopkins, I. I. Kravchenko, A. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband highly efficient dielectric metadevices for polarization control,” APL Photonics 1(3), 030801 (2016). [CrossRef]
20. Y. Guo, M. Pu, Z. Zhao, Y. Wang, J. Jin, P. Gao, X. Li, X. Ma, and X. Luo, “Merging Geometric Phase and Plasmon Retardation Phase in Continuously Shaped Metasurfaces for Arbitrary Orbital Angular Momentum Generation,” ACS Photonics 3(11), 2022–2029 (2016). [CrossRef]
21. X. Luo, “Principles of electromagnetic waves in metasurfaces,” Sci. China: Phys., Mech. Astron. 58(9), 594201 (2015). [CrossRef]
22. F. Ding, A. Pors, and S. I. Bozhevolnyi, “Gradient metasurfaces: a review of fundamentals and applications,” Rep. Prog. Phys. 81(2), 026401 (2018). [CrossRef]
23. N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013). [CrossRef]
24. F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband High-Efficiency Half-Wave Plate: A Supercell-Based Plasmonic Metasurface Approach,” ACS Nano 9(4), 4111–4119 (2015). [CrossRef]
25. S. Jiang, X. Xiong, Y. Hu, Y. Hu, G. Ma, R. Peng, C. Sun, and M. Wang, “Controlling the Polarization State of Light with a Dispersion-Free Metastructure,” Phys. Rev. X 4(2), 021026 (2014). [CrossRef]
26. X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light: Sci. Appl. 2(4), e72 (2013). [CrossRef]
27. M. Khorasaninejad, W. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffractionlimited focusing and subwavelength resolution imaging,” Science 352(6290), 1190–1194 (2016). [CrossRef]
28. W. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10(1), 355 (2019). [CrossRef]
29. C. J. C. Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4(3), 379–440 (2012). [CrossRef]
30. 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]
31. Q. Zhao, J. Zhou, F. Zhang, and D. Lippensc, “Mie resonancebased dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009). [CrossRef]
32. S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics 7(6), 1041–1068 (2018). [CrossRef]
33. 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]
34. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). [CrossRef]
35. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4(1), 2781 (2013). [CrossRef]
36. Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018). [CrossRef]
37. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016). [CrossRef]
38. D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014). [CrossRef]
39. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
40. S. L. Boris, V. V. Nikolai, P. Ramón, and A. I. Kuznetsov, “Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index,” ACS Photonics 2(7), 993–999 (2015). [CrossRef]
41. M. Kerker, D. Wang, and G. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]
42. Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008). [CrossRef]
43. Z. Ma, S. M. Hanham, Y. Gong, and M. Hong, “All-dielectric reflective half-wave plate metasurface based on the anisotropic excitation of electric and magnetic dipole resonances,” Opt. Lett. 43(4), 911–914 (2018). [CrossRef]
