Phyllotaxis-inspired Nanosieves with Multiplexed Orbital Angular Momentum

Nanophotonic platforms such as metasurfaces, achieving arbitrary phase profiles within ultrathin thickness, emerge as miniaturized, ultracompact and kaleidoscopic optical vortex generators. However, it is often required to segment or interleave independent subarray metasurfaces to multiplex optical vortices in a single nano device, which in turn affects the compactness and channel capacity of the device. Here, inspired by phyllotaxis patterns in pine cones and sunflowers, we theoretically prove and experimentally report that multiple optical vortices can be produced in a single compact phyllotaxis nanosieve, both in free space and on a chip, where one metaatom may contribute to many vortices simultaneously. The time resolved dynamics of on chip interference wavefronts between multiple plasmonic vortices was revealed by ultrafast time-resolved photoemission electron microscopy. Our nature inspired optical vortex generator would facilitate various vortex related optical applications, including structured wavefront shaping, free space and plasmonic vortices, and high capacity information metaphotonics.

responsible for one specific topological charge. Meanwhile, sufficient distance is required between meta-atoms to suppress cross coupling, which in turn degrades the device's compactness and channel capacity. To deal with this problem, one may search therapies/solutions from either frequency domain or space domain. Previously, Elhanan Maguid et. al 21 introduced asymmetric harmonic response geometric phase metasurface which realized OAM multiplexing through superposition of different harmonic components in the momentum space. In this work, we realize both free-space and near-field OAM multiplexing based on structure degeneracy in the space domain.

Results
Intriguingly, planar spirals such as Archimedean spirals 31 , logarithmic spirals 32 and Fermat spirals 33 can generate photonic OAMs with helical phase fronts. Among the Fermat spirals, the Vogel spiral 34 , also known as the "golden ratio" spiral, has been frequently studied for its unique growing pattern [35][36][37] . The pattern of a Vogel spiral 34 in polar coordinates can be described as = √ , = • 137.5°. Here, is the ordering number of a floret, c is a scaling constant, r is the radial distance between the n th floret and the center of the capitulum, is the angle between the reference direction and the position vector of the n th floret, and 137.5° is the "golden angle". The Vogel spiral is well known as one of the phyllotaxis geometries in nature, which exists in many plants including pine cones, sunflower seeds, and so on 35 . As shown in Figure 1a, multiple sets of clockwise and anti-clockwise spirals can be encoded from such a phyllotaxis geometry pattern. And the numbers of spiral arms contained in different sets are in coincidence with the Fibonacci numbers.
Such interesting phenomenon naturally arouses our interest to investigate the link between the nature-inspired pattern and optical vortices. In order to solve this puzzle, we simulated the diffraction pattern of a "golden-ratio" Vogel spiral nanosieve (c = 2.5, n starts from 1, ends at 936) in its Fresnel region at z = 300μm upon 633nm light's incidence. Indeed, the OAM spectrum analysis 38 reveals that the diffracted pattern of such a mask contains a series of OAM modes (Figure 1a), in coincidence with the numbers of spiral arms which can be encoded from the pattern. Hence, we infer that phyllotaxis-alike patterns concealing multiple spiral structures may enable the creation and multiplexing of OVs. Such beauty in nature inspires us to design phyllotaxis-alike nanosieves which can generate beams containing multiple OAM modes for both free-space and on-chip optical systems.

Design concept of phyllotaxis-inspired nanosieves
First, we revisit the "vortex comb" phenomenon 39 to obtain the working principle of our phyllotaxis-inspired nanosieves and extend it to both free-space and on-chip optical systems.
Under such circumstances, light emitted from each subwavelength nanohole of nanosieves can be approximated as a point source 40,41 . Considering a total of M point sources arranged along the azimuthal domain with equal angular separation, light emitted from the a th nanohole at plane-wave incidence can be decomposed into the summation of a set of orthogonal LG modes 42 : In equation (1), the LG modes are written in the form of , ( , ) ( ), where , ( ) denotes the complex amplitude of the corresponding LG mode, and , denotes the expanded coefficient. denotes the radial distance of the point source to the original point and z represents the focal distance. In free space, > 0, and = 0 as long as the incidence is plane wave; while in the on-chip optical system, can be approximated as zero, and = ±1 for right-(RCP) and left-handed circularly polarization (LCP) states, owing to the appropriate spinto-orbital conversion mechanism. Therefore, the final field distribution can be approximated as the interference of such M point sources, which is given by the summation of these individual elementary waves: is the summation of a finite geometric series and can be easily calculated as: Combining equations (2) and (3), we can conclude the following: For the free-space optical system, the interference pattern of such M point sources can be expressed as: while for an on-chip optical system, the interference pattern of such M point sources can be expressed as: where = 0, ±1, ±2, ±3, … in equations (3), (4), and (5).
In brief, we remark two important conclusions from the above theoretical discussion. First, multiple orders of OAM modes can be generated both in free space and in the near field via a single nanosieve device, as visible in equations (4) and (5). The appearance of these sequential OAM orders is deeply rooted in the rearrangement of the nanoholes into different sets of spirals (see various colored spiral lines in Fig. 1a and more details in the following designs), and those spirals will render the corresponding different OAMs. This is also the reason of the emerged Fibonacci sequential OAM orders embedded in a "golden-ratio" phyllotaxis nanosieve, inspired of which we call our compact devices phyllotaxis-inspired vortex nanosieves. Second, in free space, the OAM orders are independent of incident spins; while in the on-chip optical system, we can get a series of OAM modes containing spin-to-orbit conversion. Intrinsically, the surface plasmon polariton (SPP) wave excited via the circular-shape nanohole by circularly polarized light has different initial phases along different propagating directions. However, in the on-chip optical system, only SPP wave propagating towards the center of the nanosieve will interfere and form the vortices. Therefore, under circularly polarization illumination, our phyllotaxisinspired vortex nanosieve realizes spin-to-orbit conversion.

Free-space phyllotaxis-inspired vortex nanosieve
We employed the Fermat spiral with the formulation 33 = √ 0 2 + 2 0 • 2 , ( 0 ≪ 0 ), to generate a beam with tailored OAM modes in the free-space optical system. Here, denotes the azimuthal angle of the spiral, denotes the spiral radius corresponding to azimuthal angle , and 0 is the starting radius of the spiral. Therefore, light penetrating the spiral slit structure will form a helical wavefront and accumulate a • 2 phase difference on the designed focal distance 0 . Combining our previous derivation along with inspiration from the "golden ratio" phyllotaxis nanosieve, we repeated such spiral structure equally along the azimuthal angular domain l times and segmented the spiral slit structure into azimuthal equally separated nanoholes to obtain a phyllotaxis-inspired vortex nanosieve. Specifically, we choose = 633nm, 0 = 22μm, 0 = 250μm and = 13 . Here, we vary spiral azimuthal angle covering from 0 to 3π. Each of the 13 spirals is azimuthal equally segmented into 72 nanoholes.
As indicated by our theoretical insight, we now prove that our phyllotaxis-inspired vortex nanosieve can generate multiple OAMs beyond the topological charge of l. As the location of each nanohole is fixed, we can re-unite or re-sample the nanohole arrays. If we "string" the neighboring nanoholes following different trajectories, different spiral patterns can be encoded.
As is shown in the right panel of Figure 1b, four obvious sets of motifs can be encoded from the free-space phyllotaxis-inspired vortex nanosieve, which are 13 clockwise spirals, 39 anticlockwise spirals, 52 clockwise spirals, and 91 anti-clockwise spirals correspondingly. Based on this, we can infer that light coming from our free-space phyllotaxis-inspired vortex nanosieve will simultaneously carry four OAM modes with different helical wavefronts, with the correspond topological charge of l=+13, -39, +52, and -91, whose numerical amplitude intensity profile is also shown the right panel of Figure 1b.
To verify our analysis, both numerical simulation and experiments were carried out. Figure   2a and 2b shows the simulated intensity and phase profiles of the diffraction pattern of the free-space phyllotaxis-inspired vortex nanosieve at z=250 μ m upon 633 nm's illuminance respectively. It can be clearly observed that the generated on-axis four OAM patterns in Figure   2a are the superposition of the four simulated modes shown in Figure 1b with nearly no distortion. Meanwhile, one can directly obtain the modes' information from the corresponding phase profile. The free-space phyllotaxis-inspired vortex nanosieve sample was fabricated using focused-ion beam (FIB) technique on a 120-nm thick Au film above a glass substrate.
The radius of each milled nanohole is 1μm. Figure 2c shows  Since the TR-PEEM is combined with a Ti:Sapphire laser system operating at 800 nm central wavelength, the plasmonic wavelength changes to λspp ≈ 780 nm and we adjusted our design accordingly. An top-view SEM image of the sample is provided in Figure 4b. We tested the sample using the TR-PEEM system under both RCP and LCP incidences, and the relevant videos are uploaded as Supplementary Information. We have summarized the snapshots from the TR-PEEM results under RCP incidence which represent the three main stages of the vortex in dynamics formation, which are formation, revolution, and decay. Figure   4c~4e are the raw data from the TR-PEEM results. Figure 4f~4h show processed images that correspond to the delay times used in

Conclusions
In conclusion, inspired by the "golden-ratio" spiral phyllotaxis-inspired pattern in nature, we presented the idea of using phyllotaxis-inspired vortex nanosieves to generate optical beams in Nanobuilder, respectively.
For TR-PEEM measurements, we chemically synthesize single crystalline gold platelets 50 on n-doped silicon substrates, which have lateral dimensions of up to one hundred micrometers.
We then structure these atomically flat surfaces with phyllotaxis-inspired patterns, using a focused beam of Au++ ions, which is generated by a Raith IonLine plus system.
Sample characterization.

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