Abstract
For the semicircular plasmonic lens, the spiral phase is the origin of the spindependent surface plasmon polariton (SPP) focusing. By counterbalancing the spindependent spiral phase with another spiral phase or PancharatnamBerry phase, we realized the SPP focusing independent from the spin states of the excitation light. Analyses based on both HuygensFresnel principle for SPPs and numerical simulations prove that the position, intensity, and profile of the SPP focuses are exactly the same for different spin states. Moreover, the spinindependent SPP focusing is immune from the change of the radius, the central angle, and the shape of the semicircular slit. This study not only further reveals the mechanism of spindependent SPP devices but also provides effective approaches to overcome the influence of spin states on the SPPs field.
Introduction
In the threedimensional (3D) free space, optical lenses play an indispensable role in molding the flow of light, such as focusing, imaging, and optical Fourier transform (FT). However, the inherent limitations of conventional lenses are also gradually unveiled. Due to the diffraction of light, the transversal full width at half maximum of a focus is no less than about half a wavelength λ/(2n sin α), which hinders the realization of superresolution lithography and microscopy [1,2,3]. As for the optical FT relation between the front and back focal planes, the speed of the transformation is restricted by the thickness and focal length of the lens [4]. Above all, compared with the wavelength of light, the volume of the lens is bulky because of the curved surface used to achieve gradual phase accumulation [5,6,7]. And that is incompatible with the increasing demand for miniature and integrated optical devices in research and applications [8,9,10].
Surface plasmon polaritons (SPPs) which are hybrid modes of phonons and electronic oscillations propagating along the two dimensional (2D) metal/dielectric interface can be an effective tool to overcome the above limitations [11,12,13,14,15,16,17]. With the subwavelength feature, SPPs can be easily focused to a subwavelength spot [18,19,20,21]. As the counterpart of the optical lens in the 3D space, semicircular slit plasmonic lens cannot only focus SPP fields but also perform SPP FT with a much faster speed in a 2D plane [4]. Besides, in order to effectively excite SPPs, the width of the slit is smaller than the wavelength of incident light. Nevertheless, the focusing of SPPs generated by the semicircular slit strongly depends on the spin states of the incident light [22,23,24,25]. For left circularly polarized (LCP) and right circularly polarized (RCP) incident light, the focal spots of SPPs will experience spindependent transverse shifts, which is distinctive from the focusing of circularly polarized light in the free space. Since the study of the spindependent semicircular SPPs lens in 2008 by Hasman et al. [22,23,24], various mechanisms have been proposed to accomplish the spindependent SPP focusing [26,27,28]. The basic principle relies on the spindependent phase distribution accomplished by steering the orientation angles of subwavelength slits. Moreover, spindependent SPP excitation [29], SPP vortex [30], SPP hologram [31], SPP Bessel beam [32], and SPP Airy beam [33] have been demonstrated. Overall, spindependent SPP devices have been extensively studied. It is obvious and normal that the spin states of excitation light can influence the functionality of SPP devices because even the SPPs excited by a single subwavelength slit or hole depend on the spin states [24, 26, 28, 33]. However, on the contrary, is it possible to avoid the influence of spin states on the SPPs field and make the SPPs lens spinindependent?
The SPPs generated by a semicircular slit are imprinted with a spindependent spiral phase exp(iσ_{±}θ), where the spin states σ_{±} = ± 1 represent LCP and RCP light, respectively [22,23,24,25]. In this paper, we propose a global approach and a local approach to eliminate the influence of the spiral phase and achieve spinindependent SPPs focusing. The global approach deals with the semicircular slit wholly and cancels out the spiral phase by adding an opposite semicircular slit which can introduce an inversed spiral phase. Regarding the semicircular slit as the constitution of subwavelength slits, the spiral phase can be counterbalanced locally with PancharatnamBerry phase which is tuned by changing the orientation angle of the slit. The spinindependent SPP focusing is analyzed and verified with the HuygensFresnel principle for SPPs as well as numerical simulations. The robustness of the proposed approaches is tested by changing the radius, central angle, and shape of the semicircular slit. Compared with previous spindependent SPP devices [26,27,28,29,30,31,32,33], the focusing of SPPs here is independent from the spin states of the excitation light, which could improve the stability of the SPP lens.
Results and Discussions
SpinIndependent Plasmonic Lens Consisted of Double Semicircular Slits
For semicircular slit plasmonic lens illuminated by left circularly polarized (LCP) and right circularly polarized (RCP) incident light, the spiral phases increase from 0 to π counterclockwise and clockwise, respectively, as schematically shown in Fig. 1b. The spiral phase results from the interaction between the circularly polarized light and anisotropic nanoscale structure [23]. Circularly polarized light is the synthesis of the horizontally polarized and vertically polarized light with a π/2 phase difference. The SPPs excited by the two linear components can be expressed as sinθ and cosθ, respectively [25]. Thus, the SPP field generated by circularly polarized light is sinθ + exp(iσ_{±}π/2) cos θ = exp(iσ_{±}θ). Without the spiral phase, the wavefront of SPPs would be parallel to the semicircular slit and the SPP wavevector k_{sp} would be along the radial direction. However, the spiral phase corresponds to a spiral wavefront and the SPP wavevector will deviate from the radial direction, illustrated by the red and blue arrows in Fig. 1a. And, ultimately, the spiral phase results in the transverse shift of the SPP focus [22, 23, 25]. It is obvious that the spindependent spiral phase, which is the origin of the spin controlled SPP focusing, needs to be eliminated to realize the spinindependent SPP lens.
Adding another semicircular slit to introduce additional spiral phase could be a solution. When the two semicircular slits are on the same side, the two spiral phases cannot cancel each other out. Thus, the semicircular slit should be added on the opposite side. Figure 1c schematically shows the structure of the SPP lens consisted of two semicircular slits with different radius r_{1} and r_{2}. The excited SPP fields along the left and right semicircular slits can be correspondingly expressed as:
There exists a π phase difference between the spiral phases generated by two semicircular slits. Particularly, when the radiuses satisfy Δr = r_{1} − r_{2} = λ_{sp}/2, k_{sp}Δr = π could just compensate the π phase difference between the two spiral phases. As presented in Fig. 1d, the corresponding phase of SPPs is central symmetry. Concretely, the phase of SPPs generated from the point A_{1} is the same as the phase of SPPs generated from the symmetrical point A_{2}. And the SPPs generated by A_{1} and A_{2} will interfere constructively in the center, so do the other points along the semicircular slits. Accordingly, the SPPs generated by the two semicircular slits will be focused in the center without transverse shift. When the spin states of the incident light are changed, the left and right spiral phases will be reversed simultaneously and remain to be central symmetry. Therefore, the SPPs excited by both LCP and RCP light can be focused in the center of the semicircular, which indicates the spinindependent feature of the plasmonic lens.
The performance of the spinindependent plasmonic lens is analytically examined with the HuygensFresnel principle for SPPs [34, 35]. In the polar coordinate system, the SPP fields generated by the left and right semicircular slits can be respectively expressed as:
where φ denotes the angle between the radial direction and the SPP propagating path and d is the distance from the secondary source to an arbitrary point F, as shown in Fig. 1b. Substituting Eq. (1) and Eq. (2) into Eq. (3) and Eq. (4), the SPP field distributions can be obtained and are given in Fig. 2a–d. The white dashed semicircle represents the semicircular slit, and the horizontal dashed line is drawn to clearly show the transverse shift of SPPs focus. It can be seen that the direction of the transverse shift of the SPP focus is always opposite for the left and right semicircular slits. For the spinindependent plasmonic lens, the SPP distribution is the superposition of the SPP fields generated by two semicircular slits, which can be written as \( {E}_{\mathrm{sp}}\left(\rho, \theta \right)={E}_{\mathrm{sp}}^{\mathrm{L}}\left(\rho, \theta \right)+{E}_{\mathrm{sp}}^{\mathrm{R}}\left(\rho, \theta \right) \). Thus, the intensity of SPPs in the center is
where the phase difference is ΔΦ_{sp} = k_{sp}(r_{1} − r_{2}) − π and the term π results from the difference between the left and right spiral phases. To realize spinindependent focusing, the SPPs should interfere constructively in the center. Thus, the radiuses of the slits should satisfy
As presented in Fig. 2e and f, the SPP fields generated by LCP and RCP light are all focused in the center. The wavelength of incident light is 632.8 nm, and the corresponding wavelength of the SPPs λ_{sp} is 606 nm for the Au/air interface [12, 36]. The radiuses of the left and right semicircular slits are 5 μm and 4.697 μm. The normalized transversal and longitudinal distributions of the SPP focuses are extracted and compared in Fig. 2g and h. The spindependent transversal shifts of the SPP focuses in Fig. 2a–d disappear. The positions as well as the profiles of the SPP focuses generated by LCP and RCP light are exactly the same, which verifies the feasibility of the spinindependent plasmonic lens.
Fullwave numerical simulations are also performed based on the finitedifferent timedomain (FDTD) method. The parameters are kept to be the same as the ones used in the analytical calculation with the HuygensFresnel principle. The simulated SPP distributions in Fig. 3a and b agree very well with the analytical results. The transversal and longitudinal distributions in Fig. 3c and d show that the full widths at half maximum (FWHM) of the focuses along the x and ydirection (190 nm and 260 nm) are all smaller than half a wavelength. The position, the FWHM, and the intensity of the SPP focuses are all independent from the spin states of the incident light. The SPPs excited by the semicircular slits will gradually attenuate during propagation. The propagation loss is caused by the absorption in the metal [11, 12] and has been taken into consideration in the simulations by using a complex permittivity (ε_{Au} = − 11.821 + 1.426i). Thus, the propagation loss does not affect the spindependent focusing of the SPPs. Figure 3 e and f give the phase distributions around the focal spot. As indicated by the green dotted arrows, two spiral phases with clockwise and counterclockwise directions counterbalance each other, which lead to the spinindependent SPP focusing. The flat phase in the center corresponds to the focusing area. It should be noted that the phase distributions of SPPs in Fig. 3e and f are different under different spin states of the excitation light. But they are central symmetry, which requires that the intensity distributions of SPPs should be center symmetry as well. To satisfy the center symmetry requirement, the SPP focuses generated by LCP and RCP light should both be located in the center. Thus, the spinindependent intensity distributions do not necessarily mean the phase distributions are spinindependent. Here, we mainly refer to the field intensity when saying spinindependent.
The evolutions of the SPP distribution with the difference of radiuses Δr are revealed. When the radiuses satisfy Δr = nλ_{sp}, the two semicircular slits are equivalent to one circular slit with spiral phase varying from 0 to 2π. Taking Δr = λ_{sp} as an example, the spindependent SPP vortexes can be obtained, as presented in Fig. 4a and b. The phase distributions in the insets of Fig. 4a and b show that the topological charge of SPP vortexes is l = 1 and l = − 1 for LCP and RCP light, respectively. Thus, the separation Δr between the two semicircular slits has a great influence on the performance of the plasmonic lens. The two spiral phases can cancel each other out, and spinindependent SPP focusing can be accomplished only when Eq. (6) is satisfied. Moreover, according to Eq. (6), the radius and the central angle of the slits could not affect the focusing property of the plasmonic lens. For arc slits with a central angle 2π/3, r_{1} =3.7 μm and r_{2} =2.2 μm, \( \Delta r=\frac{5}{2}{\lambda}_{\mathrm{sp}} \), and the SPPs excited by LCP and RCP light are all focused in the center, as shown in Fig. 4c and d. Furthermore, the proposed approach can be applied to the spiral slits. For a spiral slit described by \( {r}_1\left(\theta \right)={r}_0+\frac{\theta }{\pi }{\lambda}_{\mathrm{sp}} \), adding another spiral slit with r_{2} = r_{1} − λ_{sp}/2 can counterbalance the spiral phase and realize spinindependent SPP focusing. The SPP distributions in Fig. 4e and f demonstrate the versatility and robustness of the proposed approach.
SpinIndependent SPP Focusing Based on PancharatnamBerry Phase
In the above discussions, we have treated the semicircular slit as a whole. As shown in Fig. 5a, a semicircular slit can be divided into subwavelength rectangle slits. In this way, the geometry PancharatnamBerry (PB) phase determined by the orientation angle of the slit is brought in [37, 38], which can be expressed as φ_{PB} = σ_{m}α. Thus, the phase of SPPs generated by each subwavelength slit is:
The spiral phase can be canceled out locally by steering the PB phase distribution. In Fig. 5a, the PB phase is a constant φ_{PB} = π/2 and has no effect on the spiral phase. When the PB phase satisfies φ_{PB} = σ_{m}θ, the spiral phase is counterbalanced locally and the phase of SPPs generated by each slit is Φ_{sp}(θ) = 0. Thus, the subwavelength slits should be aligned along the vertical direction, as shown in Fig. 5b and c.
The intensity distributions of SPPs generated by the spinindependent plasmonic lens consisted of vertical subwavelength slits are given in Fig. 6a and b. The width and length of the slits are 50 nm and 200 nm, respectively. The longitudinal and transversal profiles of the SPP focuses in Fig. 6c and d show that the position, the FWHM, and the intensity of the SPP focuses generated by the LCP and RCP light are indistinguishable. Compared with the SPP distributions in Fig. 3c and d, the transversal FWHM of the focus is about the same, while the longitudinal FWHM is more than three times larger. That is because SPPs generated by the opposite semicircular slit in Fig. 3c and d can effectively compress the transverse size of the SPP focus. Figure 6 e and f present uniform angular phase distributions around the focus, and no spiral phase is observed. That is because the spiral phase has been locally canceled by the PB phase. This is clearly different from the double semicircular slits approach which still preserves the spiral phases in Fig. 3e and f. The change of radius and central angle will not affect the focusing property of the SPP lens. Figure 6 g and h show the spinindependent SPP distributions generated by slits with a central angle 2π/3 and radius r = 2 μm.
Conclusions
In conclusion, counterbalancing the spindependent spiral phase by introducing another spiral phase or PancharatnamBerry phase is the fundamental principle of spinindependent SPP focusing. The positions and profiles of SPP focuses generated by LCP and RCP light are exactly the same with the spinindependent plasmonic lens. This study further reveals that the spiral phase is a decisive factor in determining the focusing property of the semicircular plasmonic lens. Moreover, the proposed methods can be utilized to design polarizationindependent devices in other frequency bands [39, 40] by scaling the structure.
Methods
3D numerical simulations are performed with the commercial software Lumerical FDTD Solutions. In the simulation, semicircular slits with a width of 240 nm are etched on the 150nmthick gold film and the substrate is SiO_{2} with a refractive index of 1.46. The refractive index of the gold film can be obtained from the Johnson and Christy model [36]. The mesh accuracy is set as 3, and the corresponding size of each mesh cell is about 13 × 13 × 40 nm, which can achieve a good tradeoff between accuracy, memory requirements, and simulation time. Perfectly matched layers (PML) with eight numbers of layers in the x, y, and zdirections are utilized as the boundary conditions to absorb the propagating SPP fields. Horizontally polarized light and vertically polarized light with a phase different σ_{±}π/2 are utilized to synthesize the LCP and RCP light sources. And the light source illuminates the sample from the backside to avoid its influence on the excited SPPs. To obtain the profiles of SPP focus, a 2D field monitor is placed 50 nm above the gold film, which is within the decay length of SPPs.
Abbreviations
 FDTD:

Finitedifferent timedomain
 FT:

Fourier transform
 FWHM:

Full widths at half maximum
 LCP light:

Left circularly polarized light
 PB phase:

PancharatnamBerry phase
 RCP light:

Right circularly polarized light
 SHE:

Spin Hall effect
 SPPs:

Surface plasmon polaritons
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (NSFC) (11704231, 11804199) and China Postdoctoral Science Foundation (2017 M622252).
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The refractive index of gold is obtained from Ref. [36].
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This manuscript is written by SW and YQS. The simulation is carried out by YQS, SW, and GQL. The analysis and discussion of these obtained results are carried out by YQS, GQL, and SW. All authors read and approved the final manuscript.
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Li, G., Sun, Y. & Wang, S. SpinIndependent Plasmonic Lens. Nanoscale Res Lett 14, 156 (2019). https://doi.org/10.1186/s1167101929902
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DOI: https://doi.org/10.1186/s1167101929902