1. Introduction

Recently, optical spin angular momentum (SAM) and orbital angular momentum (OAM) [1, 2] have been extendedly studied in lots of optical research fields, such as optical communication [3, 4], metasurface [5–8], optical data storage and encryption, for theirs great potential in these fields. The notable difference between these two momenta is that SAM depends on the polarization of beam but OAM arises from the helical phase fronts whose Poynting vectors are not parallel to the beam. SAM has the value of $\pm \hslash$ per photon. It means that light is right- or left-circularly polarizatized. However, a beam which carries OAM equal to $\pm \ell \hslash$ per photon ($\ell$ is an integer) shows a helical wavefront with the topological charge of $\pm \ell .$ In free space, when OAM interferes with a plane wave or a spherical wave, the superposition wave projecting on the screen will show a spiral pattern. Most of researches use complex systems or unique metasurface arrays [9–20] to generate OAM in free space.

The OAMs in confined metallic waveguide structures have also drawn a lot of attention for producing the spiral surface plasmon (SSP) modes in the waveguide. For example, the Archimedean spiral grating that is carved on a metal film can generate OAMs and optical vortices of SP [9,12–17]. They can be applied in trapping or rotation of dielectric microparticles and circular polarization transmission filters. On the other hand, the SSP modes propagating inside metallic nanoholes and on metallic nanorods also own the OAMs [4,21–23]. These SSP modes can be generated by incident nanorod’s (nanohole’s) eigenmodes onto the end of nanorod (nanohole). When the SSP is transferred to emitting light at the other end of nanorod or nanohole, it can propagate a short distance and preserve the handedness. However, how to extract the OAM of SSP modes from metallic nanorods and transform it into an orbital motion of SP mode have never been considered. To generate a spiral SP pattern on a concentric metal ring grating has never been proposed and examined, either.

In this study, a concentric Ag ring grating placed under an Ag nanorod with SSP modes on the nanorod is investigated by FDTD simulations and theoretical analyses. The SSP modes on the nanorod are generated by normally incident nanorod’s eigenmodes onto the end of the nanorod. The ring grating placed under the nanorod extracts the OAM of SSP and transforms it into the orbital motion of SP on the grating. The orbital motion of SP on the grating forms a plasmonic Archimedean spiral pattern. The mechanism of formation of Archimedean spiral modes is elucidated. The relation between the topological charges of SSP mode on the nanorod and the spiral patterns of SP on the grating is also clarified.

2. Simulation structure and method

Figure 1(a) plots the simulation model. It is divided into two parts: an Ag nanorod and an Ag concentric ring grating (which is formed by carving grooves on a square Ag substrate). The grating is placed under the nanorod (the axes of nanorod and ring grating are aligned with each other). The detailed geometric parameters in the simulations are listed in Table 1. Notably, the concentric ring grating in this study can be fabricated by a commercial focused ion beam system. The nanorod and grating are embedded in air. The wavelength of incident light in free space (${\lambda }_0$) is 633 nm. The FDTD program MEEP is utilized in the simulation [24]. The three-dimensional x-y-z coordinate system is adopted (the axes of nanorod and ring grating are along the z-direction). The dimensions of grid cells in x, y and z directions are also given in Table 1. The surrounding boundaries of the model are perfectly matched layers. The Drude model of Ag is used with the plasma frequency and collision frequency set to $1.32\times {10}^{16}$ rad/s and $0.68\times {10}^{14}$ rad/s, respectively [25].

 

Fig. 1 (a) Simulation model and (b) schematic for SSP mode on Ag nanorod coupling onto Ag ring grating.

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Table 1. Geometric parameters for the simulation.

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To excite single-stranded (triple-stranded) SSP modes on nanorod, the TM₀ and ${\text{HE}}_{\pm \text{1}}$ (HE₁ and HE_-2) eignemodes are normally incident onto the end face of nanorod [22,23]. In the simulation, the TM₀ eigenmode is excited by normally incident the Ez-polarized plane wave onto the end face of nanorod. The ${\text{HE}}_{\pm \text{1}}$ eigenmodes are excited by combining an E_x-polarized plane wave and an E_y-polarized plane wave with a phase difference of $\text{π}/2$ between them and also normally incident onto the end face of nanorod. To excite the HE_-2 eigenmode, two circularly polarized plane waves that are modulated by the space functions $\cos \theta$ and $\sin \theta$ ($\theta$ is the azimuth angle in the cylindrical coordinate system), respectively, with a phase difference of $\text{π}/2$ between them are also normally incident onto the end face of nanorod.

3. Results and discussion

The origin of Archimedean spiral pattern on the Ag ring grating is analyzed first. The wavevector (k) of SSP mode on Ag nanorod can be divided into vertical ($k_{\perp }$) and horizontal ($k_{//}$) components (see Fig. 1(b)). The horizontal component $k_{//}$ can be further divided into $k_{r,rod}$ and $k_{\theta ,rod}$ components. When the SSP mode of Ag nanorod couples into SP on Ag grating, the r-directional wavevector (k_r) of SP on Ag gratings can be expressed as

After the SP on Ag grating is induced, it will propagate on the grating with the initial phase velocities of ${\upsilon }_r$ and ${\upsilon }_{\theta }.$ Moreover, the phase velocities of SP are assumed to be unchanged under the condition of modest loss. Therefore, the SP on Ag ring grating will exhibit the Archimedean spiral pattern (which is a spiral with constant radial and angular velocities) and is described by the following polar coordinate (r, $\theta$) equation

The above mentioned Archimedean spiral patterns are also examined by FDTD simulations. Figures 2(a) and 2(b) plot the simulated time-averaged power density contours at the top surface of grating for the period of Ag gratings equal to 80 nm and the topological charges of SSP modes on Ag nanorod equal to −1 and 1, respectively. (The topological charges of −1 and 1 correspond to the left- and right-handed, respectively, single-stranded SSP modes on Ag nanorod. And the handedness of SSP modes is defined by the rotation direction of the spiral pattern along the propagation direction (i.e. -z direction in Fig. 1(a) [22].) (The electric fields of Archimedean spiral modes have both ${\text{E}}_{\text{//}}$ (E_x and E_y) and ${\text{E}}_{\perp }$ (E_z) components.) Figs. 2(a) and 2(b) display typical Archimedean spiral patterns. Furthermore, the values of $r_0$ and P in Eq. (4) can be extracted from Figs. 2(a) and 2(b). Thus the Archimedean spiral patterns can be predicted by Eq. (4). The white lines in Figs. 2(a) and 2(b) present the predicted spiral patterns, which agree with the simulation results. The small discrepancy between the predicted Archimedean spirals and simulated power density contours in Figs. 2(a) and 2(b) should come from the phenomenon of plasmonic Bloch oscillation (PBO) in cylindrical metal–dielectric waveguide arrays [26] (i.e. the cylindrical Ag ring grating in this work). When SP propagates along $\theta$-direction in the waveguide arrays, PBO will cause the SP alternatively to move toward inner layers and outer layers of the waveguide arrays and finally induce an error in the extracted value of P in Eq. (4). (It is worth to mention that SP on the ring grating does not carry the optical OAM anymore.)

 

Fig. 2 Simulation results of time-averaged power density contours at the top surface of grating for topological charge $\ell$ equal to (a) −1 and (b) 1. $\text{Λ}$ = 80 nm. The geometrical parameters are listed in Table 1.

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The mechanism of formation of Archimedean spiral pattern mentioned above can be further verified by studying the relation between $\text{Λ}$ (period of grating) and P (period of Archimedean spiral). Taking the time differential of Eq. (4), the relation between ${\upsilon }_r$ and ${\upsilon }_{\theta }$ is expressed as

 

Fig. 3 (a)–(c) Simulation results of time-averaged power density contours at the top surface of grating for periods of ring grating equal to 40 nm, 60nm and 120 nm, respectively. (d) Simulation results (red triangle) and linearly least-squared fitting curve (black line) of $\text{1/Λ}$ versus 1/P. The topological charge $\ell$ is −1. ${\lambda }_0$ = 633 nm.

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Subsequently, the spiral SP mode on Ag ring grating that comes from the SSP mode on Ag nanorod with the topological charge $\ell =3$ is also investigated. The geometry parameters for this investigation are also listed in Table 1. Here we consider that the SSP mode is composed of HE₁ and HE_-2 eigenmodes (i.e. the SSP mode is triple-stranded [22,23]). (Thus, the topological charge of SSP mode is 3 [1 - (−2) = 3]). Figure 4 presents the simulated time-averaged power density contours at the top surface of grating ($\text{Λ}$ = 80 nm). Figure 4 displays a triple-stranded Archimedean spiral pattern on the grating. The predicted spiral patterns using Eq. (4) are also plotted in Fig. 4 for comparison. The simulated and predicted patterns areconsistent with each other. Figure 4 reveals that the OAM of SSP on Ag nanorod is transferred into the orbital motion of SP on the Ag ring grating. The number of strands of Archimedean spiral pattern matches the topological charge (i.e. $\ell$) of SSP mode. It should be noted that the proposed structure and excitation can be used to shape the near-field pattern with special symmetry. For example, the clover-shaped power intensity pattern near the center of the ring grating for the topological charge $\ell =3$ (Fig. 4) has the C₃-symmetry. It is more convenient to explore C₃-symmetry molecules and metamaterials.

 

Fig. 4 Simulated time-averaged power density contours (at the top surface of grating) with three-stranded Archimedean spiral pattern for topological charge $\ell =3$. The geometrical parameters are listed in Table 1. ${\lambda }_0$ = 633nm.

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Finally, the SSP mode on Ag nanorod coupling onto the straight Ag grating is investigated. Figure 5(a) plots the simulated structure. In Fig. 5(a), the period of gratings ($\text{Λ}$) is equal to 80 nm. The width of the central grating under the nanorod (gw) is 120 nm. (See Table 1). Because the $\theta$-directional velocity of SP on the straight grating cannot be maintained, the spiral pattern will disappear. Figure 5(b) presents the simulated time-averaged power density contours at the top surface of grating for SSP mode on Ag nanorod with $\ell =-1$ (which is composed of TM₀ and HE_-1 modes). Figure 5(b) indicates that the energy of SSP mode is coupled onto the right grating. However, the spiral pattern vanishes as discussed above. When $\ell$ is changed from −1 to 1, the energy of SSP mode will be coupled onto the left grating. These results can be applied to design switch or logical devices. Similarly, because the $\theta$- and r-directional velocities of SP cannot remain unchanged on a planar Ag plate, the spiral pattern will not form on the plate.

 

Fig. 5 (a) Simulation model for SSP mode on Ag nanorod coupling onto Ag straight grating. (b) Simulated time-averaged power density contours at the top surface of grating for topological charge $\ell =-1$. The geometrical parameters are listed in Table 1. ${\lambda }_0$ = 633 nm.

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The propagation lengths of SP modes (defined as the lengths after which the intensities decrease to 1/e) on the Ag ring grating and straight grating are also calculated in this work. The propagation lengths for the single-stranded (Figs. 2(a) and 2(b)) and triple-stranded (Fig. 4) Archimedean spiral modes on ring grating are the arc lengths of SP rotating $0.25\text{π}$ radians and $0.15\text{π}$ radians, respectively, from its initial position. The propagation length for SP on the straight grating (Fig. 5(b)) is 322 nm (four times the grating period). The thickness of Ag substrate is set as 200 nm in the simulations. When the thickness of substrate is reduced to 30 nm, the Archimedean spiral patterns still appear on the grating. Furthermore, our simulation results show that these spiral patterns can also be observed at the bottom surface of the substrate with a substantial reduction in power density. From the experiment viewpoint, the more convenient way to observe the Archimedean spiral patterns is to irradiate the nanorod placed on one side of the grating substrate and to image the power density using NSOM on the other side of the substrate. Here we reemphasize the difference between this work and the previous studies [9,12–17]. In previous researches, the spiral gratings are used for SP acquiring OAM and generating vortex. Conversely, this work demonstrates that a concentric ring grating extracts the OAM of SSP and transforms it into an orbital motion of SP which is a constant-velocity motion and hence displays an Archimedean spiral pattern.

4. Conclusion

The plasmonic Archimedean spiral modes that exist on concentric Ag ring gratings are observed. These modes are generated by placing the ring grating under an Ag nanorod to extract the OAM of SSP on the nanorod and transform it into the orbital motion of SP on the grating. When SSP on nanorod are coupled into SP on ring grating, both the r- and $\theta$-directional wavevectors are conserved. Furthermore, both the r- and $\theta$-directional velocities of SP keep unchanged when it propagates on the ring grating. The formation of Archimedean spiral patterns is ascribed to these two factors. The linear relationship between 1/P and $\text{1/Λ}$ is also observed in FDTD simulation, which further confirms the mechanism of formation of Archimedean spiral pattern. In addition, the number of strands of Archimedean spiral pattern matches the topological charge of SSP mode. When the ring grating is replaced by the straight grating, the energy of SSP on nanorod is still coupled into SP on grating. However, the spiral pattern vanishes because the $\theta$-directional velocity of SP is not conserved. The Archimedean spiral SP modes have potential applications in the fields of data storage, dielectric micro-particle manipulation, biosensing and directional switching.