## Abstract

Taking advantage of the phase modulation of surface plasmon polaritons (SPPs) generated by subwavelength slits, we elaborately designed an axicon-shaped slit array to dynamically manipulate the profiles of Bessel-like SPPs beams by changing the polarization of the incident light. As the polarization direction of linearly polarized light changes from horizontal to vertical, the profile of SPPs beam will evolve from the zeroth-order Bessel function to the first-order Bessel function. For circularly polarized light, the geometry Pancharatnam-Berry (PB) phase is introduced and can lead to the transverse shift of the main lobe of SPPs beam. The evolution of the beam profile and the distance of the transverse shift are theoretically illustrated by analyzing the phase distribution of SPPs. The proposed versatile approach suggests charming applications including on-chip communications, polarization-controlled particle manipulation and the design of polarization-based dynamic SPPs devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

With the unique feature of subwavelength field confinement, surface plasmon polaritons (SPPs) are promising candidate to realize photonic version of on-chip integrated circuit [1]. Benefiting from the development of micro- and nano-fabrication techniques, components of the plasmonic circuit including lenses [2,3], waveguides [4], splitters [5], multiplexers [6] and logic gates [7] have been extensively studied. However, owing to the metal absorption and the scattering by rough surface, the propagation length of SPPs is attenuated, which hinders the further application of SPPs. Moreover, the inherent diffraction nature of SPPs can lead to an extra coupling loss between different components. Thus, inspired by the nondiffracting optical beams in the free space, diffraction-free SPPs beams are introduced to minimize the transverse spread of SPPs during propagation. Besides, the self-healing property of diffraction-free SPPs beams can effectively reduce the influence of surface roughness on SPPs. Among the nondiffracting SPPs beams with the profiles of Bessel [8–10], Airy [11–13], Mathieu and Weber [14,15] function, the Bessel-like SPPs beam is more favorable because of its straight forward trajectory.

By using nonlinear materials [16] or changing the properties of incident light [17–20], SPPs distributions can be flexibly modulated and thus dynamic SPPs devices capable of higher reconfigurability and multi-functionality are achieved. The excitation of SPPs is closely related to the polarization of the incident light and only the polarization component perpendicular to the slit can effectively give rise to SPPs [20]. Therefore, modulating the polarization of incident light has been a simple and effective way to realize dynamic SPPs devices. Polarization-controlled excitation [20,21], focusing [22–25], vortex generation [26,27], hologram [28] and propagation [29] of SPPs have been realized. However, polarization has not been employed to dynamically manipulate SPPs beams yet. And most of the nondiffracting SPPs beam devices are static. Polarization-based dynamic SPPs devices have not been fully exploited.

In this paper, an axicon-shaped slit array is designed to realize polarization-controlled manipulation of Bessel-like SPPs beams. SPPs beam excited by horizontally polarized (HP) light exhibits the profile of the zeroth-order Bessel function, while SPPs beam with the profile of the first-order Bessel function is obtained for vertically polarized (VP) light. With the incident of left circularly polarized (LCP) and right circularly polarized (RCP) light, due to the presence of geometry Pancharatnam-Berry (PB) phase, the main lobe of the SPPs beam will shift downward and upward, respectively. The nondiffracting and self-healing properties of the Bessel-like SPPs beam are verified as well. Furthermore, the dependence of beam profile and the transverse shift on the polarization and the parameters of the slit array are revealed. Compared with previous studies in which Bessel-like SPPs beams with different profiles were realized by using two devices [8–10,30,31], here the desired SPPs beam profile can be readily obtained by changing the polarization of the incident beam, which hints that the dynamic Bessel-like SPPs beam in this manuscript can provide more degrees of freedom in practical applications. Moreover, the axicon-shaped slit array can effectively extend the functionality of the previous polarization-controlled SPPs devices [20–29] and may inspire the design of novel dynamic SPPs devices.

## 2. Results and discussions

To generate Bessel-like SPPs beams, subwavelength rectangular slits are designed to constitute an axicon-shaped array, as shown in Fig. 1. In analogy with the intersecting grating approach [8,10,32], the slit array consists of two (the upper and lower) subarrays which are both arranged obliquely along the y-axis with an angle $\theta $. The upper and lower subarrays are symmetrical about the x-axis and thus the orientation angles of the slits in the two subarrays are $\alpha $ and $-\alpha $ respectively. An advantage of breaking the intersecting grating into subwavelength slits is that the phase and amplitude of SPPs can be steered by changing the orientation angle of the slits [25,33,34] and thus subwavelength slits can provide more degrees of freedom to manipulate SPPs. With the normal incidence of light, SPPs generated by each slit will propagate and interference in the xy plane and SPPs beams with Bessel profile can be constructed in the central area along the positive x direction. For SPPs generated by each slit, the amplitude is the same due to the identical absolute value of orientation angles of the slits in the two subarrays, however, the phase is different. Besides the phase shift $\phi \left(r\right)$ induced by the different optical paths, an additional geometry PB phase $\phi \left(\alpha \right)$ can be imprinted on the SPPs excited by circularly polarized light because of the different orientation angles of the slits ($\alpha $ and $-\alpha $). Therefore, the phase of SPPs can be expressed as ${\Phi}_{sp}=\phi \left(r\right)+\phi \left(\alpha \right)$. Compared with the intersecting grating approach where only phase $\phi \left(r\right)$ is involved, the PB phase $\phi \left(\alpha \right)$ introduced here is polarization dependent and can be tuned by changing $\alpha $. Thus, by carefully engineering the orientation angle $\alpha $ and the polarization of the incident light, polarization-controlled dynamic manipulation of Bessel-like SPPs beams can be readily realized.

In order to reveal the dependence of SPPs beams generated by the axicon-shaped slit array on the polarization of light, we start from considering the incident of linearly polarized light. The frequency of the incident light is 0.75 THz and the corresponding wavelength is 400 μm. The oblique angles of the upper and lower subarrays are set to be $\theta =\pi /24$ and the orientation angles $\alpha $ of the corresponding slits in the two subarrays are $\pi /4$ and $-\pi /4$. The width and length of subwavelength slits are 30 μm and 200 μm, respectively. The slits are etched on the gold film with a thickness of 100 nm and the substrate is silicon with a refractive index of 3.42. The whole slit array contains 50 subwavelength slits and the longitudinal separation between the each slit is 200 μm. Numerical simulations based on 3D finite difference time domain (FDTD) are carried out using commercial software FDTD Solutions. For horizontally polarized (HP) and vertically polarized (VP) light, the profiles of the generated SPPs beams are shown in Figs. 2(a) and 2(b) respectively. It is clear that SPPs beam generated by HP light exhibits a main lobe with maxima in the centre while the main lobe vanishes and a hollow beam with minima in the centre emerges for SPPs beam generated by VP light. The normalized transversal intensity distributions along x = 8 mm which are plotted in Fig. 2(c) more clearly show that the profiles of SPPs beams generated by HP and VP light correspond to the zeroth-order and the first-order Bessel function [10], respectively. Resembling the different focusing properties (one or two focuses) of the semicircular SPPs lens illuminated with HP or VP light [2,22,35], the different SPPs beam profiles here can be illustrated by checking the phase of the generated SPPs. For HP incident light, SPPs generated by the upper subarray and the lower subarray are in phase and can interfere constructively along the positive x-axis. However, SPPs generated by the two subarrays are out of phase for VP light and thus will interfere destructively. Therefore, by changing the polarization direction of the linearly polarized light, SPPs beam with the profile of the zeroth-order or the first-order Bessel function can be selectively acquired. The efficiency of the proposed axicon-shape slit array can be obtained by calculating the ratio between the intensity of the incident light ${I}_{inc}$ and the intensity of the excited SPPs ${I}_{spp}$, which can be expressed as $\eta ={I}_{spp}/{I}_{inc}$. With the incident of HP and VP light, the efficiencies of the proposed structure are calculated to be 15.5% and 16.4%, respectively.

The transversal intensity distributions along x = 4 mm, x = 8 mm and x = 12 mm in Fig. 2(a) are extracted and plotted in Fig. 2(d) to analyze the nondiffracting feature of the Bessel-like SPPs beam. The full widths at half maximum (FWHM) of the main lobe are 733.3 μm, 810.3 μm and 725.6 μm, respectively, which quantitatively demonstrates that the diffraction of the SPPs beam is weak during the propagation. The self-healing properties of the Bessel-like beams are studied by placing an obstacle with a radius of 250 μm at x = 4 mm, which is represented by the white circle in Figs. 2(e) and 2(f). Although the SPPs field around the obstacle is affected, the Bessel profiles of the SPPs beams after the obstacle are not severely deteriorated.

With the incident of left circularly polarized (LCP) and right circularly polarized (RCP) light, the distributions of the SPPs beams are given in Figs. 3(a) and 3(b), respectively. The profiles of the SPPs beams are both similar to the zeroth-order Bessel beam, however, the main lobes deviate transversally from the centre. The transversal distributions along x = 8 mm plotted in Figs. 3(c) and 3(d) clearly show that the main lobe shifts downward for LCP light and shifts upward for RCP light. The distance of the transversal shift is about $\pm $380 μm for LCP and RCP light, respectively. The efficiencies of the axicon-shaped slit array are the same $\eta $ = 15.9% for LCP and RCP incident light.

The displacement of the main lobe may remind us of the transversal shift of the SPPs focus which results from spin-dependent geometrical phase of partial waves for semicircular SPPs lens [22,23,35]. Here, we also analyze the phase of SPPs excited by the upper subarray and the lower subarray, which can be expressed as:

where ${k}_{sp}=2\pi /{\lambda}_{sp}$ is the wave number of SPPs, ${d}_{u}=\left|\frac{{y}_{0}-{x}_{0}\cdot \mathrm{cot}\theta}{\sqrt{1+{\mathrm{cot}}^{2}\theta}}\right|$and ${d}_{l}=\left|\frac{{y}_{0}+{x}_{\text{0}}\cdot \mathrm{cot}{\theta}_{0}}{\sqrt{1+{\mathrm{cot}}^{2}\theta}}\right|$ are the distances from point $\left({x}_{0},{y}_{0}\right)$ to the upper subarray and the lower subarray, respectively, ${\phi}_{u}\left(\alpha \right)$ and ${\phi}_{l}\left(\alpha \right)$ are the PB phases induced by the interaction between circularly polarized light and the subwavelength slits. As discussed in [33], the SPPs field generated by the subwavelength slit is proportional to ${e}^{{\sigma}_{\pm}i\alpha}\mathrm{sin}\alpha $, where ${\sigma}_{\pm}=\pm 1$ represent LCP and RCP light, respectively. Thus, the PB phase follows $\phi \left(\alpha \right)={\sigma}_{\pm}\alpha $. To form the main lobe around the x-axis, the phase difference of SPPs generated by the two subarrays should satisfy $\Delta {\Phi}_{sp}={\Phi}_{sp}^{u}-{\Phi}_{sp}^{l}\text{=}0$, which meansSubstitute the PB phases ${\phi}_{u}\left(\alpha \right)={\sigma}_{\pm}\alpha $, ${\phi}_{l}\left(-\alpha \right)={\sigma}_{\pm}\left(-\alpha \right)$, ${d}_{u}$ and ${d}_{l}$ into Eq. (3), we can obtain the transversal shift of the main lobeRepresented by the green dots in Figs. 3(c) and 3(d), the theoretical value calculated with Eq. (4) is $\pm $383 μm, which is in good agreement with the simulated result.The transverse shift of the main lobe induced by the PB phase can be eliminated by breaking the symmetry of the two subarrays. As schematically shown in Fig. 4(a), the upper subarray is moved a distance of ${\lambda}_{sp}/4$ along the direction perpendicular to the subarray and the two subarrays are not symmetrical about the x-axis any more. The change of position means an additional $\pi /2$ phase is imposed on the SPPs excited by the upper subarray. The phase difference of SPPs generated by the two subarrays can be written as $\Delta {\Phi}_{sp}=-{k}_{sp}\cdot {\lambda}_{sp}/4+{\phi}_{u}\left(\alpha \right)-{\phi}_{l}\left(-\alpha \right)$. For LCP incident light, ${\phi}_{u}\left(\alpha \right)-{\phi}_{l}\left(-\alpha \right)=\pi /2$ and $\Delta {\Phi}_{sp}=0$, suggesting the constructive interference of SPPs. Thus, as shown in Fig. 4(b), the transversal shift disappears and a main lobe in the centre along the x-axis is generated, which corresponds to the zeroth-order Bessel SPPs beam. However, with the illumination RCP light, ${\phi}_{u}\left(\alpha \right)-{\phi}_{l}\left(-\alpha \right)=-\pi /2$ and $\Delta {\Phi}_{sp}=-\pi $. Consequently, SPPs beam with the profile of the first-order Bessel function is formed, which can be seen from Fig. 4(c). The transversal distributions in Fig. 4(d) more clearly exhibit the different profiles of the Bessel SPPs beams. Here, we demonstrate that Bessel-like SPPs beams with different profiles can also be realized by switching the handedness of circularly polarized light, except for changing the polarization direction of linearly polarized light.

Furthermore, as indicated by Eq. (4), it can be concluded that the transverse shift of the main lobe is determined by the handedness of the circularly polarized light, the orientation angle $\alpha $ of the slits and the oblique angle $\theta $ of the subarray. Thus, by changing either one of these properties, the transverse shift of the main lobe can be modulated. For another axicon-shaped slit array with different parameters ($\alpha \text{=}\pi /\text{6}$ and $\theta \text{=}\pi /\text{18}$), Figs. 5(a) and 5(b) show the distributions of Bessel-like SPPs beams excited by LCP and RCP light respectively. The transverse shift of the main lobe becomes smaller but can still be recognized. The transverse distributions of the SPPs beams along x = 8 mm are shown in Figs. 5(c) and 5(d), which clearly reveal the main lobe shifts downward and upward for LCP and RCP light respectively. The simulated transverse shift is about $\pm $187 μm and agrees well with the theoretical value (the green dots).

## 3. Conclusions

In conclusion, with one single device, i.e. the elaborately designed axicon-shaped slit array, we have demonstrated that the Bessel-like SPPs beams with different profiles can be selectively excited by modulating the polarization of the incident light. Concretely, the acquisition of the zeroth-order Bessel SPPs beam and the first-order Bessel SPPs beam corresponds to the incident of HP and VP light respectively, and the main lobe of the SPPs beam will shift transversally for circularly polarized light. The polarization-controlled phase modulation of SPPs achieved by the subwavelength slits plays an essential role in realizing dynamic manipulation of Bessel-like SPPs beams. It has been demonstrated that both free space vortex beam [36] and SPPs vortex [27] can be used to manipulate particles. Thus, considering that Bessel beam in the free space can function as an optical tweezers [37], polarization-controlled particle manipulation should be feasible with the dynamic Bessel-like SPPs beam. Combined with SPPs slot waveguides and SPPs logic gates [7], the axicon-shaped slit array may also find applications in on-chip communications. Moreover, the dynamic Bessel-like SPPs beam can be realized in other wavebands by scaling the parameters of the slit and structure.

## Funding

Program 973 (2013CBA01702); National Natural Science Foundation of China (NSFC) (11704231, 11647015).

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