Digitized subwavelength surface structure on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation

doi:10.21203/rs.3.rs-20639/v1

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Digitized subwavelength surface structure on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation

https://doi.org/10.21203/rs.3.rs-20639/v1

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Optical vortices carrying orbital angular momentum (OAM) have recently attracted increasing interest for providing an additional degree of freedom for capacity scaling in optical communications. The optical vortex generator is an essential component to facilitate OAM-enabled optical communications. Traditional devices face challenges of limited compactness, narrow bandwidth and first-order OAM modes. Here, using the direct-binary search (DBS) optimization algorithm, we design, fabricate and demonstrate a digitized subwavelength surface structure on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation. It features an ultra-compact footprint (~3.6×3.6 μm 2 ) and ultra-wide bandwidth (1480-1630 nm), supporting two polarizations and high-order OAM modes (OAM +1 , OAM -1 , OAM +2 , OAM -2 ) with high purity of ~90%. The mode crosstalk matrix is measured in the experiment with favorable performance. When generating x-pol. OAM +1 , x-pol. OAM -1 , y-pol. OAM +1 and y-pol. OAM -1 , the crosstalk of the worst case is less than -14 dB. When generating OAM +1 , OAM -1 , OAM +2 and OAM -2 , the crosstalk between any two OAM modes is less than -10 dB, and the lowest crosstalk is about -17 dB. The wavelength-/polarization-/charge-diverse optical vortex generator enables the full access of multiple physical dimensions (wavelength, polarization, space) of lightwaves. The demonstrations may open up new perspectives for chip-scale solutions to multi-dimensional multiplexing optical communications.

Photonics/optics

Integrated optics devices

subwavelength structures

optical vortices

optical communications

Light typically consists of a stream of linearly polarized photons, carrying a linear momentum equivalent to ħk per photon (ħ: reduced Plank’s constant; k: wavenumber) when traveling in a straight line. Light travelling through space may rotate, so light can also carry angular momentum. There are two typical ways in which the light can rotate, therein the circular polarization rotation is a manifestation of photon spin, referred to the spin angular momentum (SAM). A spin angular momentum of σħ is carried by a photon of energy ħω, where ω is the angular frequency and σ = 1 or − 1 for left- and right-handed circular polarizations, respectively. Beyond that, the light can also rotate with its spatial phase structure, leading to a specified helical phase front and carrying additional angular momentum, namely the orbital angular momentum (OAM) [1]. Generally, a light beam carrying OAM, equivalent to a value of lħ per photon (l: topological charge), is called optical vortex with an annular intensity structure incorporated with an azimuthal phase structure of exp(ilϕ), where ϕ is the azimuthal angle. The topological charge or mode order l, in principle, can take any integer value, positive or negative. Such distinct features of optical vortex have fueled multitudinous research and applications in many branches of optical physics, including optical manipulation, optical tweezers, optical spanners, particle trapping, memory, microscopy, imaging, sensing, metrology, astronomy, nonlinear interaction, and quantum science [2–14]. Very recently, OAM-carrying optical vortex has also attracted increasing interest in optical communications [15–20]. Remarkably, being fully compatible with existing multiplexing resources such as time, wavelength and polarization in optical communications, OAM as a subset of spatial modes provides an additional physical dimension for multiplexing, which allows the continuous scaling of transmission capacity and spectral efficiency. OAM beams have helical phase fronts with the number of intertwined helices and the direction depending on the magnitude and the sign of l, respectively. The topological charge l, also referring to the mode order, can take an unbounded value theoretically, so an infinite number of OAM beams could be accommodated provided an infinitely large aperture. Even for a finite aperture in a practical communication system, multiple OAM beams can still be used to assure the capacity scaling by OAM multiplexing. The space-division multiplexing (SDM) exploiting OAM mode basis, provides a promising approach to achieve large capacity beyond traditional multiplexing techniques for free-space, underwater and fiber optical communications [15–40].

The generation of OAM-carrying optical vortices is essential for almost all the OAM-enabled applications, including OAM-based optical communications. Unlike ordinary light beams with a planar phasefront, the generation of optical vortices characterized by doughnut intensity profiles and helical phasefronts usually relies on bulky devices used for mode conversion, such as cylindrical lens pairs, spiral phase plates, diffractive optical elements, and spatial light modulators [41–44]. Additionally, resonator cavities, q-plates, metamaterials and metasurfaces are also used for generating optical vortices [45–52]. Very recently, for the sake of miniaturization and integration, photonic integrated devices have been demonstrated to generate optical vortices [53–65]. In particular, silicon photonic integration platform plays an increasingly important role because of its miniaturization, reliability, scalability and complementary metal-oxide semiconductor (CMOS) compatibility for low-cost mass production [66, 67]. The previous works based on silicon photonic platform such as waveguide phase arrays, whispering-gallery-mode resonators and holographic grating, succeeded in generating optical vortices with impressive performance. The challenges remain, especially limited compactness, narrow bandwidth and first-order OAM modes. Remarkably, the full compatibility of OAM multiplexing with existing well-established multiplexing techniques such as wavelength-division multiplexing (WDM) and polarization-division multiplexing (PDM), is essential to OAM-assisted SDM optical communications. In this scenario, ultra-compact nanophotonic devices on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation are of great research value. The wavelength-/polarization-/charge-diversity can facilitate OAM multiplexing with high-order optical vortices and its compatibility with WDM (ultra-broadband property) and PDM techniques.

In this paper, based on a nonlinear search algorithm called direct-binary search (DBS), we propose, design, simulate, fabricate and demonstrate a digitized subwavelength surface structure on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation with outstanding performance. Such nanostructured silicon photonic device enables the generation of optical vortices with two OAM values (OAM_+ 1, OAM_− 1) and two polarization states (x-/y-polarization) in a wide wavelength range from 1480 to 1630 nm. Moreover, diverse optical vortex generation beyond the first-order OAM is also available, e.g. ultra-broadband and polarization-diversity generation of optical vortices with different OAM values (OAM_+ 1, OAM_− 1, OAM_+ 2, OAM_− 2). The mode crosstalk performance is also characterized in the experiment.

Firstly, we design a chip-scale optical vortex generator used for the generation of x-pol. OAM_+ 1, x-pol. OAM_− 1, y-pol. OAM_+ 1 and y-pol. OAM_− 1. As illustrated in Fig. 1(a) and (b), the fundamental TE0 mode respectively from four input ports in different directions is coupled into the vertically emitted optical vortices with specific topological charge value and polarization state after propagation through the 2D digitized subwavelength surface structure on silicon platform. It should be noted that the optical vortex is mainly characterized by the 3D spiral phase structure, thus the generation of optical vortices primarily depends on the phase modulation of the incident in-plane guided mode (fundamental TE₀ mode), which could be realized by a subwavelength surface structure (holographic fork grating). Based on this method depending on the modulation of the propagation phase delay, OAM_+ 1 and OAM_− 1 can be generated with high purity. However, significant challenge remains to further improve the quality of the generated optical vortices and generate high-order optical vortices, where more accurate phase modulation during in-plane guided mode to free-space optical vortex conversion is highly desired. Here, localized phase modulation is introduced by further digitizing the conventional holographic fork grating into a more general subwavelength surface structure. Shown in Fig. 1(a) is the zoom-in details of the proposed digitized subwavelength surface structure of the optical vortex generator, which is designed and fabricated on a standard silicon-on-insulator (SOI) platform with a normal 220 nm silicon layer and 2 µm buried oxide layer. The designed digitized nanostructure region is shallow etched down normally 60 nm with a compact footprint of 3.2 × 3.2 µm², discretized by numerous 100 nm × 100 nm silicon/silicon dioxide pillars, namely ‘pixels’. Thus we call it digitized subwavelength surface structure on silicon platform.

This type of digitized nanophotonic structure has been increasing popular to design photonic integrated devices with arbitrary topologies used for multiple novel functionalities [68–76]. Unlike conventional design techniques with only a small amount of parameters optimized one by one, the digitized design based on optimization algorithms allows the full parameter space searching of possible structures even without user intervention, which could be seen as a promising technical innovation. Recently, a crowd of powerful optimization strategies have been developed for nanophotonic design, ranging from heuristic optimization methods to topological design method. Several classical heuristic optimization methods, such as genetic algorithm, simulated annealing algorithm, particle swarm algorithm and attractor selection algorithm, have traditionally been applied to many fields. Critically, most of them are built on a somewhat limited parameterization of the solution space and requires abundant plentiful random testing of different parameter sets, putting the application in relatively complicated models under high computational cost. To avoid the test of a large number of possible solutions before finding a satisfactory one, an inverse design method is introduced under the inspiration of the variable density method used in continuum topology optimization [72–76]. The device structure could be discretely parametrized into a 2D image with each pixel corresponding to the value of the permittivity. In general, the optimization procedure could be decomposed into two stages. Firstly, the permittivity of each pixel is allowed to vary continuously between that of silicon and silicon oxide, where the gradient descent method is normally used to generate a well preformed initial structure. Then a binary discrete constraint is imposed on the permittivity of them because the fabrication material is only chosen to be silicon or silicon oxide.

Although many basic integrated devices have been designed with remarkable performance and compact footprint in this way, the drawback is reasonably obvious as well. Above all, a good initial condition is of crucial importance to the entire optimization design process. As mentioned, the continuous optimization stage is essentially relied on the descent gradient calculated with precise full wave electromagnetic simulation, which will still consume a lot of computing resources even with the adjoint method [77, 78]. Besides, the continuous parameter is normally ended with weekly modulated permittivity in between, leading to severe degradation of device performance after determining the final binary discrete structure, and this problem could not be solved in essence with the recently proposed simple approximation or the level set method [79]. In this work, we get around this problem by defining the material of each pixel to be digitized binary at first, and the size is chosen to be 100 nm × 100 nm, totally satisfying requirements of the manufacturing process. Then the design is followed by the DBS optimization algorithm. Originally the initial structure is generated randomly by MATLAB, and the optimization turns out to be sensitive to the starting designs, hardly to get an optimal design. Then comes the question how to generate a well performed initial structure without the continuous stage. As stated above, modulation of the propagation phase could be realized by a surface holographic fork grating [64]. Hence, an initial structure of holographic fork grating is chosen, which could be used to generate polarization diversity first-order OAM modes. Next, the pixels are chosen to switch its state in a random order, and a figure-of-merit (FOM) is calculated each time for comparison. The FOM is defined as the average phase purity calculated for all emitted OAM modes coupled from different input ports of the device. The pixel state is retained once the FOM is improved, if not, the state is reversed and the algorithm proceeds to the next pixel. One iteration could iterate over all the pixels and more iterations will be executed until no more improvement exhibits on the FOM. In the end, the final structure is numerically simulated with three-dimensional finite-difference time-domain (3D-FDTD) method to prove the validity of the designed digitized subwavelength surface structure on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation.

In Fig. 2(a) we give the detailed simulation results of the generated wavelength-/polarization-/charge-diverse OAM modes with two mode orders and two polarization states ranging from 1480 to 1630 nm, respectively. The intensity profile of the generated OAM modes is firstly studied with an annular structure, whose far field is calculated by the scalar diffraction theory. To evaluate the quality of the generated OAM modes more accurately, we further study the mode purity of them, so that their interferogram and phase distribution are given as well. The interferogram is obtained via the interference between the generated OAM mode and a coherent reference Gaussian beam, in which the number and hand of spirals corresponding to the order info including the magnitude and sign of the OAM modes. Besides, one can tell the mode order and evaluate the phase purity from the extracted phase distribution. Remarkably, based on the synthesis of a series of digitized subwavelength nanostructures with low Q resonance, the designed OAM generator exhibits well-deserved broadband characteristic. Hence, the property characterization of the generated OAM modes turns out to be basically identical in the simulation in the C-band. Even within a wider wavelength range, little degradation happens to the calculated purity of the generated OAM modes. As shown in Fig. 2(c), all x-pol. OAM_+ 1, x-pol. OAM_− 1, y-pol. OAM_+ 1 and y-pol. OAM_− 1 modes are generated with high purity (> 0.88) when varying the wavelength from 1480 to 1640 nm. Considering that four OAM modes vary slightly from each other and share the same tendency of wavelength-dependent purity, one can see that the average purity reaches a maximum value of 0.93 around 1550 nm, which is designed to be the central wavelength. We compare the performance between the nanostructure after DBS optimization and the initial discrete structure of holographic fork grating. Taking a glance at the average purity value of the generated OAM modes in Fig. 2(e), one can find out an obvious improvement above 10% in the mode purity using the nanostructure after DBS optimization. It should be noted that the initial structure prepared for iterative optimization is discretized by digitized binary from the ideal holographic fork grating structure. A performance degradation would be brought into in this discrete process inevitably, so it is necessary to introduce the localized phase control. Followed by the DBS iterative optimization, it turns out the OAM modes could be generated with even better quality assisted by the more precise phase control.

In order to fully exploit the spatial dimension of lightwaves in OAM-assisted SDM optical communications, high-order OAM modes beyond OAM_+ 1 and OAM_− 1 are also expected to be emitted from an OAM generator. Based on the same DBS iterative optimization method, we further study a charge diversity optical vortex generator supporting high-order OAM modes. The schematic illustration of the high-order OAM generator is shown in Fig. 1(b). With light incident from four input ports, OAM_+ 1, OAM_− 1, OAM_+ 2 and OAM_− 2 modes could be generated. Following the same design principle, what differs from the first one is that the designed nanostructure region contains 36 × 36 pixels shallowly etched down 60 nm with each pixel size of 100 nm × 100 nm. The number of pixels is expanded slightly, which could give rise to a relatively more refined phase control to generate high-order OAM modes.

From the simulation results shown in Fig. 2(b), one can clearly identify the characteristics of the generated four charge diversity OAM modes (OAM_+ 1, OAM_− 1, OAM_+ 2, OAM_− 2). Slight nonuniform distribution is observed in the far-field intensity profile, which could be ascribed to the inhomogeneous distribution of digitized nanostructured pixels, each differing in emission efficiency when the in-plane guided mode propagating through the nanostructured region. Instead of the helical phase distribution, more extra factors on the intensity distribution could be considered in the global optimization to avoid this fact. From the simulated interferograms and phase distributions, one can easily tell the OAM mode not only from the number and direction of the spirals, but also from the phase change (-2π, 2π, -4π, 4π) along the azimuthal direction. The calculated mode purity exhibits slight difference between the first-order OAM mode and the second-order OAM mode, as shown in Fig. 2(d). Although it is reasonable that the purity of OAM_± 2 mode is slightly lower than the OAM_± 1 mode, OAM modes with the same |l| but opposite sign also slightly differ in purity. This could be ascribed to the fact that the judgment condition of FOM only considers the average mode purity of all generated OAM modes. Thus, the maximum purity of the OAM_+ 1 mode is 0.96 while it is 0.93 for the OAM_− 1 mode. The purity of the OAM_+ 2 mode could reach 0.92 while OAM_− 2 mode 0.9. The purity of the OAM_+ 1 mode and OAM_− 1 mode is larger than 0.94 and 0.91, respectively, within the wavelength range from 1480 to 1630 nm. On the other side, the purity of the OAM_+ 2 mode and OAM_− 2 mode slightly decreases from 0.89 to 0.84 with the increase of the wavelength from 1480 to 1630 nm.

Based on the optimized design of the digitized subwavelength surface structure proved by simulations, we fabricate the chip-scale optical vortex generator on an SOI wafer with a 220 nm thick top silicon layer and a 2 µm thick buried oxide layer. In the fabrication process, firstly the electron-beam-lithography (EBL) is employed to define the layout pattern, then an inductively coupled plasma (ICP) etching is followed to etch the top silicon layer. Waveguide outlines are etched fully down 220 nm to the buried oxide, while gratings are shallow etched down normally 60 nm. After that, a 1-µm-thick SiO₂ layer is deposited on the top by plasma enhanced chemical vapor deposition (PECVD), covering the whole device as the upper cladding. Figure 3(a) and 3(b) depict the measured microphotograph of the fabricated polarization diversity optical vortex generator with x-pol. OAM_+ 1, x-pol. OAM_− 1, y-pol. OAM_+ 1 and y-pol. OAM_− 1 mode, where the digitized subwavelength surface nanostructure region is connected by four adiabatic tapers from different directions for emitting specific OAM modes, respectively. The high-order optical vortex generator with OAM_+ 1, OAM_− 1, OAM_+ 2 and OAM_− 2 mode is fabricated with the similar digitized subwavelength surface structure, whose microphotograph is shown in Fig. 3(c) and 3(d).

Shown in Fig. 4(a) is the measured far-field intensity profiles and corresponding interferograms of the generated x-pol. OAM_+ 1, x-pol. OAM_− 1, y-pol. OAM_+ 1 and y-pol. OAM_− 1 ranging from 1480 to 1620 nm, respectively. Agreed with simulations, the measured intensity profiles display a doughnut shape and the interferograms with spiral interference patterns indicate the helical phasefront of the generated OAM modes. It is noted that the sign of the topological charge is indicated by the chirality of the interference pattern. Besides, only slight degeneration of the mode quality is observed towards longer wavelength. Using the same experimental setup, we then experimentally study the wavelength-/polarization-/charge-diverse generation of high-order optical vortices by another device. The far-filed intensity profiles and interferograms are measured as shown in Fig. 4(b). The first-order OAM modes (OAM_+ 1, OAM_− 1) exhibit well-performed intensity and phase distributions in a wide wavelength range from 1480 to 1630 nm. On the other hand, the second-order OAM modes (OAM_+ 2, OAM_− 2) show slight performance degradation in the intensity profiles, while the magnitude and sign of the OAM order can still be deduced from the measured interferograms within a wide wavelength range. The measured intensity profiles and interferograms shown in Fig. 4 confirm the successful generation of wavelength-/polarization-/charge-diverse optical vortices with favorable performance, i.e. OAM_+ 1, OAM_− 1, OAM_+ 2 and OAM_− 2 with two polarizations in a wide wavelength range (1480–1630 nm).

The proposed chip-scale wavelength-/polarization-/charge-diverse optical vortex generator is expected to be used in multi-dimensional multiplexing optical communications, i.e. combined OAM multiplexing, WDM and PDM. Hence, the crosstalk between different OAM modes is of great importance. Here, an experimental measurement method based on a spatial light modulator (SLM) is used to measure the mode crosstalk matrix. The experimental configuration is shown in Fig. 3(f). The optical vortex emitted from the chip is demodulated by a polarizer (polarization demodulation) and an SLM loaded with a holographic phase pattern (OAM demodulation). Objective lens and convex lens are used to adjust the beam size projected onto the SLM. Note that the SLM is sensitive to polarization, so a polarizer and a half-wave plate are placed in the light path to adjust the light polarization to be aligned to the working direction of the SLM. The polarizer is set to be x- or y-direction for polarization demodulation and two kinds of pure phase patterns are selectively loaded onto the SLM to demodulate the OAM_+ 1 or OAM_− 1 mode into Gaussian-like beam, respectively. Hence, as shown in Fig. 5(a), the first column shows the far-field OAM intensity profiles without using a demodulation phase pattern. When a corresponding phase pattern is loaded onto the SLM, unlike OAM modes with doughnut shape intensity profile, the demodulated light after removing the helical phasefront displays a bright spot at the beam center, just like a Gaussian beam. The results with both a correct polarization demodulation and a correct OAM demodulation are shown in the diagonal elements (bright spots at the beam center) in Fig. 5(a). By contrast, as shown in the off-diagonal elements in Fig. 5(a), it is noted that the light beam remains a doughnut shape when using a non-corresponding demodulation phase pattern, while almost no light could be observed when setting the polarizer to an orthogonal direction. Hence, the mode crosstalk matrix measurement is followed by filtering and coupling only the central part of the demodulated beams into fiber connected with an optical power meter. According to the recorded mode crosstalk matrix in Fig. 5(b) and 5(c), a crosstalk less than − 14 dB is measured at the worst case. Similar measurement is carried out for the crosstalk between OAM_+ 1, OAM_− 1, OAM_+ 2 and OAM_− 2 modes generated by the high-order optical vortex generator, where holographic phase patterns for the demodulation of OAM_+ 1, OAM_− 1, OAM_+ 2 and OAM_− 2 modes are loaded onto the SLM instead. In the same way, light displays a bright spot at the beam center with a correct demodulation (diagonal), while a non-corresponding demodulation (off-diagonal) only results in null intensity at the beam center. The measured intensity distribution matrix is shown in Fig. 5(d). The recorded mode crosstalk matrix is displayed in Fig. 5(e) and 5(f). The crosstalk between any two OAM modes is less than − 10 dB, and the lowest crosstalk is about − 17 dB. The experimental results shown in Figs. 4 and 5, especially the relatively low crosstalk between the wavelength-/polarization-/charge-diverse OAM modes, indicate the high-performance optical vortex generator and its great potential for multi-dimensional multiplexing optical communications (OAM multiplexing + WDM + PDM).

The demonstrated wavelength-/polarization-/charge-diverse optical vortex generator based on digitized subwavelength surface structure shows outstanding performance. Some future perspectives towards a more robust optical vortex chip could be considered as follows

1) The nonlinear optimization algorithms might be further improved to design the silicon-based optical vortex chip more efficiently with superior performance in terms of mode purity and mode crosstalk.

2) The basic concept of digitized subwavelength surface structure might be expanded to realize wavelength-/polarization-/charge-diverse optical vortex generator with even more input ports (> 4) so that more OAM modes are achievable using a single chip.

3) High-efficient and low-crosstalk coupling and packaging of the fabricated optical vortex chip with fibers might be developed for practical applications, i.e. coupling and packaging of input ports with SMFs and vertical emission output with OAM fiber.

4) The silicon-based optical vortex chip might be used both as the emitter and detector. In particular, assisted by the fiber-chip-fiber coupling and packaging, the proposed wavelength-/polarization-/charge-diverse optical vortex chip might be employed in practical multi-dimensional multiplexing fiber optical communications for capacity scaling.

5) The digitized subwavelength surface structure on silicon platform might be further developed as a more generalized design method to realize much more complicated and robust functional photonic chip.

In summary, we propose and demonstrate CMOS-compatible ultra-compact integrated wavelength-/polarization-/charge-diverse optical vortex generator with a broadband wavelength range from 1480 and 1630 nm, supporting two polarizations and high-order OAM modes. Based on the designed digitized subwavelength surface structure optimized with the DBS algorithms, x-pol. OAM_+ 1, x-pol. OAM_− 1, y-pol. OAM_+ 1 and y-pol. OAM_− 1 mode could be generated with the mode purity exceeding 90%. Moreover, the generation of high-order OAM modes including OAM_+ 1, OAM_− 1, OAM_+ 2, and OAM_− 2 could be also realized with the same design method. We fabricate and comprehensively characterize the chip-scale wavelength-/polarization-/charge-diverse optical vortex generator. Outstanding performance is achieved in terms of mode purity and mode crosstalk. For the generation of x-pol. OAM_+ 1, x-pol. OAM_− 1, y-pol. OAM_+ 1 and y-pol. OAM_− 1 mode, a crosstalk less than − 14 dB is obtained at the worst case. For the generation of OAM_+ 1, OAM_− 1, OAM_+ 2 and OAM_− 2 mode, the crosstalk between any two OAM modes is less than − 10 dB, and the lowest crosstalk is about − 17 dB. The demonstrated wavelength-/polarization-/charge-diverse optical vortex generator provides the possibility of fully accessing multiple physical dimensions of lightwaves such as wavelength, polarization and space. The demonstrations open a way to implement multi-dimensional multiplexing optical communications with capacity scaling, i.e. incorporating compatible ultra-compact chip-enabled OAM multiplexing with well-established WDM and PDM techniques.

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Availability of data and materials

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Competing interest

The authors declare that they have no competing interests.

Funding

This work was supported by the National Key R&D Program of China (2019YFB2203604); the National Natural Science Foundation of China (NSFC) (11774116); the Natural Science Foundation of Hubei Province of China (2018CFA048); the Key R&D Program of Guangdong Province (2018B030325002); the Program for HUST Academic Frontier Youth Team (2016QYTD05); and the Fundamental Research Funds for the Central Universities (2019kfyRCPY037).

Authors' contributions

J.W. developed the concept. J.W. and X.P.C. designed the photonic integrated devices and conceived the experiments. N.Z., S.Z. and X.P.C. carried out the experiments. J.W. and X.P.C. performed the theoretical analyses, simulations, acquired the experimental data, and carried out the data analyses. S.G., Y.Z., M.H. and X.C. fabricated the devices. X.C. and Y.Z. provided the technical supports in the experiments. J.W. and X.P.C. contributed to writing the paper. J.W. finalized the paper. J.W. supervised the project.

Acknowledgements

Not applicable.

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Digitized subwavelength surface structure on silicon platform for wavelength-/polarization-/charge-diverse optical vortex generation

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