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.