Controllable plasmonic vortex sequence with on-chip discrete-slit-based metalens

Like free-space vortex beams, surface plasmon polaritons can carry orbital angular momentum to form plasmonic vortices (PVs). Recently, research interest in PV fundamentals and applications has increased. However, generating and manipulating the topological charges of PVs over wide ranges using on-chip devices remains challenging. Here, we propose an on-chip plasmonic metalens structure to generate tunable PV sequence with controllable topological charges at discrete wavelengths. When compared with conventional spiral-slit structures, the designed metalens has additional structural parameters that bring more degrees of freedom to control the range and interval of the topological charge distribution of the PV sequence. Analytical and simulation methods are used to verify the metalens’ functionality. It is proved that the topological charges of the generated PV sequence are symmetrically distributed about the fundamental mode (l = 0), which cannot be realized by a traditional Archimedean helix. In addition, the normalized powers of the PV sequence are all above 0.8, showing that the designed metalens structure has potential for use as an on-chip optical vortex comb device. This work has potential applications in on-chip optical information processing, integrated optical communications, and optical tweezers.


Introduction
Surface plasmon polaritons (SPPs) are evanescent electromagnetic waves that propagate along metaldielectric interfaces, and SPPs have shown great application values in super-resolution imaging [1] optical storage [2,3], optical sensing [4], Raman scattering [5], nano-lithography [6], and plasmonic waveguide devices [7,8]. Recently, their ability to carry surface-confined orbital angular momentum (OAM) and form plasmonic vortices (PVs) [9][10][11][12][13] has drawn widespread interest, with studies of the spatiotemporal dynamics of PVs via photoemission electron microscopy [14] and spin-orbit coupling during PV formation [15]. Because of OAM, PVs are able to trap and rotate particles along circular trajectories, including nanoscale objects that cannot be controlled precisely using conventional optical tweezers; PVs thus have a wide range of applications in optical trapping and manipulations [16][17][18]. In addition, PVs can be used to generate and control optical skyrmions and have important applications in information storage and spintronics devices [19,20].
To develop PV applications further, it will be highly important to study PV generation and manipulation methods, particularly in for compact and integrated optical systems. At present, two main methods are used to generate PVs. The first uses an optical vortex (OV) directly as the incident beam to excite PVs, which usually requires large-sized optical devices such as spiral phase plates, gratings and spatial light modulators. Therefore, it is unsuitable for integrated optical systems. The second method uses on-chip metallic nanostructures, which provides very compact device sizes for integrated optical systems. For example, Archimedean spiral slits etched on gold layers have been used widely to generate PVs [21][22][23][24][25][26], where the use of spiral slits lead to the generation of a spiral phase for the excited SPPs and finally the formation of a PV at the center. Very recently, metasurface has been listed as a superior candidate for manipulating light arbitrarily [27][28][29], many more complex on-chip plasmonic metasurfaces have been designed to generate various SPP distributions, further increasing the degrees of freedom for PVs generation and manipulations [30,31]. For example, a plasmonic metasurface composed of an Archimedean spiral distribution of cross units can excite spin-dependent PVs at the center of its structure [32], and a metasurface composed of multiple rectangular nanoslits enables generation of higher order SPP vortices [33]. In addition, plasmonic metasurfaces can also be used to generate spin-independent PVs [34,35], and PVs with asymmetric intensity distributions at the centers of their structures or change the PV generation location [36,37]. However, in these on-chip plasmonic nanostructures, it is difficult to control the topological charge of the generated PVs freely. Although it is possible to increase or decrease the topological charge by one through the spin-orbit coupling effect [15], modulation of the topological charge over a wide range remains challenging.
In this paper, we propose a new PV generation and manipulation method based on an on-chip plasmonic metalens structure composed of discrete arc-shaped nanoslits. When compared with the traditional Archimedean spiral-slit structures [14][15][16][17][18], our designed metalens has more adjustable parameters for modulation of the topological charge of the generated PV over a wide range, and the generated topological charge is symmetrically distributed about the fundamental mode (l = 0). The metalens structure is studied via both analytical modeling and Lumerical finite-difference time-domain (FDTD) simulations. Using this structure, a PV sequence with topological charge l that varies with wavelength can be generated, and both the range and interval of the topological charge distributions of the PV sequence can be manipulated freely using several key parameters. As a result, a continuous change of topological charge from 10 to −10 can be achieved over the 700-900 nm wavelengths. In addition, the normalized powers are all above 0.8 at the wavelengths corresponding to the integer topological charges, showing that the designed structure has potential for use as an on-chip OV comb device.

Design of the on-chip metalens
As shown in figure 1(a), a radially polarized broad-band light source is normally incident from the bottom to the metalens structure. After SPP excitation by the metalens, a PV sequence can be generated at the center with different topological charges at discrete wavelengths, and the topological charges of the generated PVs are symmetrical about the fundamental mode (l = 0). The metalens is shown in detail in figure 1(b) and consists of a series of arc-slits (slit width: 400 nm) etched on a gold film on a glass substrate. The thickness of gold film mainly influences the direct transmittance of incident light and the excitation efficiency of SPP. The directly transmitted light could have an adverse effect on the excited PV field when the thickness of gold film is very thin, while the thick gold film may decrease the SPP excitation efficiency. Thus, according to previous experimental research [38], the gold film thickness is selected as 200 nm. The number of arc-slits is N, where each arc-slit has the same angle of 2π/N but a different radius. The radius of the nth arc-slit satisfies the following relationship: where R n is the radius of the n-th arc-slit, R 1 is the radius of the first arc-slit, and D is the radius difference between adjacent arc-slits. In our design, D = mλ SPPC , where m is an integer and λ SPPC is the plasmonic wavelength corresponding to a selected incident central wavelength λ C . For λ C , the phase difference between SPPs excited using two adjacent arc-slits as determined by the radius difference D = mλ SPPC is always m × 2π, and thus the designed structure shown in figure 1(b) with discrete arc-slits is equivalent to a circle slit (or a plasmonic lens [39,40]) for the excited SPP. As a result, for a radially-polarized incident light at λ C , PV cannot be formed because of the phase difference of m × 2π between adjacent arc-slits (equivalent to zero phase difference), instead, a bright SPP focal spot is formed at the center. However, at other wavelengths, the phase difference between two adjacent arcs is no longer m × 2π, and the relative phase at another λ (phase difference between λ and λ C ) corresponding to the n-th arc-slit can be calculated as: where is the SPP wavelength for the incident wavelength λ, and ε m and ε d are the dielectric constants of gold and air, respectively. The dielectric constant of the gold film at different wavelengths was calculated using the Drude model [41]. The phase difference between the first and the final arc-slits (ϕ(N, λ) − ϕ(1, λ)) determines the final topological charge of the PV formed at the wavelength λ. Therefore, the topological charge generated using the metalens structure can be calculated as: .
Using equation (3), we select the parameters N = 9, D = λ SPPC ≈ 767 nm, (λ C = 785 nm), R 1 = 20 µm (R n varies from 20 µm to 26.136 µm), and calculate the topological charges of PVs generated within the 700-900 nm wavelength range, as indicated by the black line in figure 1(c). The figure shows that l = 0 is generated at λ C = 785 nm, as expected, and l = 1 and l = −1 are generated at wavelengths of λ = 704 nm and λ = 891 nm, respectively. For comparison, we also considered the case of a traditional Archimedean spiral-slit structure (inset in figure 1(c)) with a continuous variation in radius from 20 µm to 26.136 µm (same as the metalens structure), and find that the topological charge l is changed from l = 6 to 8 within the same wavelength range, as indicated by the red line in figure 1(c). This occurs because the topological charge distribution in the Archimedean spiral-slit structure is determined solely by the distance between the head and the tail of the helix, and thus cannot be regulated further via other parameters, e.g. D and N. Moreover, its generated topological charge cannot be distributed symmetrically about the fundamental mode (l = 0) like the metalens (l = 1 to −1), as shown in figure 1(c).
To verify the result from equation (3), we calculate the dominant E z component distribution of the PV field on the gold film surface using the following equation [42]: where k SPP = 2π/λ SPP is the SPP wave vector, λ is the light source wavelength, γ is the angle between MO and MN ( figure 1(b) 2 is the distance between M (ξ , ζ) and N (x, y), and φ is the azimuth angle. The amplitude and phase distributions of the excited PV field at λ = 704 nm, 785 nm, and 891 nm calculated using equation (4) are shown in figure 1(d). Ring-shaped PV fields with l = 1 and l = 1 are shown to be generated at 704 nm and 891 nm, respectively, while a bright focal spot for the SPP appears at the central wavelength of 785 nm; these results are all consistent with the theoretical predictions in figure 1(c).
To verify the theoretical results further, numerical simulations were carried out using the FDTD (commercial code) to simulate the PV field under the same conditions. All the six boundaries for the computation volume are terminated with perfectly matched layers to avoid parasitic unphysical reflections around the structure. A special grid is set up for the metal film, the grid resolution is 10 nm along both x and y direction, and 5 nm along z direction. The FDTD results shown in figure 1(e) agree well with the analytical results in figure 1(d).
Although the calculation time of FDTD simulation takes about six times than the analytical model with the same mesh accuracy, the FDTD simulation includes the loss and edge scattering of the structure, which is closer to the real experimental conditions, hence in the following studies we only show the FDTD results.

Influence of D on the value of topological charge
To modulate the topological charges of the generated PV sequence, several main structural parameters are investigated. We first consider the effect of radius difference D on the topological charge. When the value of D is doubled from D = λ SPPC ≈ 767 nm to D = 2λ SPPC ≈ 1534 nm, with all other parameters remaining unchanged, the topological charge of the PV sequence calculated using equation (3) is shown as the black line in figure 2(a). Unlike the topological charge l changing from 1 to −1 in figure 1(c)

Influence of rotational symmetry of structure on topological charge
Next, we study the effect of the metalens structure's rotational symmetry on the topological charges of the PV sequence. We extend the structure in figure 1(b) to rotationally symmetrical structures with two, three and four orders, as shown schematically in figures 3(a)-(c), respectively. In this case, the topological charge of the generated PV can be calculated as where β is the order of rotational symmetry of the structure and the other parameters are same as those in equation (3). The FDTD simulated amplitude and phase distributions of the PV field (E z component) with β = 2, 3, and 4 are shown in figures 3(d)-(f), respectively, and the corresponding topological charge distributions at the different wavelengths are shown in figure 3(g). Comparison with the results in figure 1 shows that when the structures have rotational symmetry with two, three, and four orders, the wavelengths of the integer topological charges of the PV do not change, but the corresponding topological charge intervals are increased to 2, 3, and 4, respectively. Therefore, the topological charge interval of the PV sequence can be modulated by varying the structure's rotational symmetry.

Influence of the arc-slit number N on topological charge
Finally, we study the effect of the arc-slit number N on the topological charge of the PV sequence. According to equation (5), within the same wavelength range, a wider range of topological charges can be obtained with larger values of N. As an example, we select the double rotational symmetry structure ( figure 3(a)) and investigate the topological charge distribution of the PV via both equation (5) and FDTD simulations.  figure 4. When N is changed from N = 17 to N = 41, the topological charge range of the PV sequence can be extended greatly, from 4 to −4 up to 10 to −10, thus verifying that the topological charge range can be controlled easily using N. However, when N continues to increase, the distribution of PVs generated by the metalens may have poor quality due to the damping effect of SPP propagation [23]. To generate a larger range of topological charge, the metalens structure can be designed on a dielectric substrate with low loss [43].  Taking N = 41 as an example, we calculated the normalized power spectrum for different topological charges of the PV sequence, with results as shown in figure 5. The normalized power is defined here as the ratio of the electric field amplitudes of the different topological charges to the maximum amplitude in the x-y plane. The results show that the normalized powers are all above 0.8 at the wavelengths corresponding to the integer topological charges, verifying that the designed structure has potential for use as an on-chip OV comb device.

Conclusion
In conclusion, we propose a plasmonic on-chip metalens structure to generate tunable PV sequences with different topological charges within a given wavelength range. When compared with the conventional Archimedean spiral-slit structure, the designed metalens has more tunable structural parameters, and thus can provide more degrees of freedom to modulate the topological charges of the generated PV sequences, and the generated topological charge is symmetrically distributed about the fundamental mode (l = 0), which cannot be realized by conventional Archimedean spiral-slit structure. Three key structural parameters, comprising the radius difference D, the rotational symmetry β, and the number of arc-slits N, are studied via analytical methods and FDTD simulations. It is shown that both the ranges and the intervals of the topological charge distributions of the PV sequences can be controlled freely by varying these parameters. In theory, this method can be used to generate desired PV sequences in any given wavelength range, and it is also suitable for use with other optical surface waves, e.g. Bloch surface waves [43]. In addition, based on the spin-orbit coupling effect [44], the circularly polarized light can also be used in the metalens structure to increase or reduce the topological charge by one for the PV sequence. Although this work is studied with theoretic calculation and FDTD simulation, it can be experimentally verified. The proposed metalens can be fabricated by the electron beam lithography system like previous works [45]. The formed SPP field distribution can be measured by several methods such as a fiber tip-type near-field scanning optical microscope [9] or photoelectron emission microscopy [14]. The proposed method has potential applications in on-chip optical information processing, integrated optical communications, and optical tweezers.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).