Enlarging spin-dependent transverse displacement of surface plasmon polaritons focus

Yuqing Sun; Chunying Zhao; Guoqun Li; Xing Li; Sen Wang

doi:10.1364/OE.27.011112

1. Introduction

Spin Hall effect of light (SHEL), which results from the coupling between spin angular momentum (SAM) and orbital angular momentum (OAM), manifests itself as a spin-dependent transverse displacement perpendicular to the refractive index gradient for light refracted or reflected by the interface between homogeneous media [1–4]. In general, the transverse displacement is only dozens of nanometers due to the weak spin-orbit interaction (SOI) and thus the experimental detection usually relies on ultrasensitive quantum weak measurements [2,5] or multiple reflections [3]. Although the SHEL is a tiny phenomenon, it still holds promising important applications in optical communications [6], precision metrology [7–9] and quantum information processing [4,10]. To enhance the SOI and enlarge the displacement, various mechanisms including Brewster’s angle reflection [11], metasurface [12–14] and topological edge states [15] were adopted and giant SHEL has been achieved. Nevertheless, SHEL is mostly studied in 3D free space with bulky system, and realizing the SHEL in a more compact and integrated 2D manner without weakening the SHEL is highly desirable.

On the other hand, surface plasmon polaritons (SPPs) propagating and evanescently confined along the metal/dielectric interface provide a feasible platform to realize 2D photonic circuit [16–20]. Moreover, the SPPs generation is strongly dependent on the polarization of the excitation light [21–28]. With a semicircular slit plasmonic lens, the focus of SPPs excited by circularly polarized light will experience a spin-dependent transversal shift, which is analogous to the SHEL in the 3D free space [29–33]. In terms of SOI and Coriolis effect, the unified spiral phase brings out the spin-dependent splitting of SPPs focus and the displacement can be estimated with $δ y \approx λ_{s p} / 2 π$ [29]. In the terahertz regime, the spiral phase is experimentally obtained and the polarization-controlled focusing of SPPs is analyzed from the perspective of Fourier transformation [34]. Overall, spin-dependent displacement of SPPs focus has been extensively studied and the underlying physical mechanism has been clearly revealed. However, an obvious disadvantage is that the separation between the SPPs focuses generated by left circularly polarized (LCP) and right circularly polarized (RCP) light is on the subwavelength scale ( $\approx λ_{s p} / π$ ), which limit the potential application of semicircular plasmonic lens. And little effort has been devoted to enlarging the spin-dependent transverse displacement.

In this paper, we enlarge the spin-dependent displacement of SPPs focus by at least six times with a semicircular plasmonic lens consisted of orthogonal slit pairs. Theoretical analyses reveal the linear relationship between the SPPs phase and the orientation angle of the slit pair. Thus, arranging the slit pairs along a semicircle can introduce additional spiral phase which results in the enlargement of the transverse displacement. The dependence of the displacement SPPs focus on the total spiral phase which includes intrinsic spiral phase and the additional spiral phase is obtained. Moreover, reversed spin-dependent displacement and continuous control of the displacement of SPPs focus can be realized with the proposed mechanism.

2. Results and discussions

Before studying the spin-dependent focusing property of the semicircular plasmonic lens, it’s helpful to analyze the amplitude and phase of SPPs generated by subwavelength slits which have been the commonly used building blocks for SPPs devices [21–23,25–27,35–38]. As shown in Fig. 1(a), a rectangular slit with orientation angle $α$ is illuminated by circularly polarized light $E$ which can be decomposed into a horizontally polarized component $E_{h}$ and a vertically polarized component $E_{v}$ with a phase difference $π / 2$ . For linearly polarized incident light, the SPPs generated by the nanoslit at point F can be expressed as $E_{s p} = \frac{\sin α}{\sqrt{r}} \exp (i k_{s p} r)$ , where $r$ is the distance from the slit to the point F and $k_{s p}$ is the wave vector of SPPs [37]. Since only the polarization component perpendicular to the nanoslit can effectively excite SPPs, the SPPs generated by the $E_{h}$ component and the $E_{v}$ component are $E_{s p}^{h} = \frac{\sin^{2} α}{\sqrt{r}} \exp (i k_{s p} r)$ and $E_{s p}^{v} = \frac{\cos α \sin α}{\sqrt{r}} \exp (i k_{s p} r)$ , respectively. Thus, with circularly polarized incident light $E$ , the generated SPPs field is the superposition of the SPPs generated by the two orthogonal components and can be written as $E_{s p}^{σ_{\pm}} = E_{s p}^{h} + \exp (σ_{\pm} i π / 2) E_{s p}^{v}$ , where the different spin states $σ_{\pm} = \pm 1$ represents LCP and RCP light, respectively. After simplification, the SPPs field generated by a single slit can be written as

E_{s p}^{σ_{\pm}} = \sin α \exp (i σ_{\mp} α) \frac{\exp (i k_{s p} r + i σ_{\pm} π / 2)}{\sqrt{r}},

which indicates a sinusoidal amplitude distribution and a linear phase distribution.

Fig. 1 Schematic diagram of a subwavelength rectangular slit (a) illuminated by circularly polarized light, (b) and (c) are the amplitude and phase distribution for left circularly polarized (LCP) light and right circularly polarized (RCP) light, respetively. (d) shows an orthogonal slit pair, (e) and (f) are the corresponding amplitude and phase distribution for different orientation angles $α$ .

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The linear phase can be referred to as geometry Pancharatnam–Berry (PB) phase which is caused by the interaction between circularly polarized light and nanoslits [36]. Simulated amplitude and phase distributions based on commercial software FDTD Solutions are given in Figs. 1(b) and 1(c), and agree well with the theoretical results based on Eq. (1). For the single nanoslit, the linear PB phase is charming. However, the accompanied sinusoidal amplitude distribution may bring about inconvenience in modulating the SPPs filed. Adding another orthogonal nanoslit with orientation angle $β = α + 90^{\circ}$ could be a solution to optimize the amplitude modulation [21], which is shown in Fig. 1(d). The SPPs filed generated by the orthogonal slit pair is the superposition of the SPPs generated by each slit and can be obtained by $E_{s p}^{σ_{\pm}} = \sin α \exp (i σ_{\mp} α) \frac{\exp (i k_{s p} r + i σ_{\pm} π / 2)}{\sqrt{r}} + \sin β \exp (i σ_{\mp} β + i k_{s p} d) \frac{\exp (i k_{s p} r + i σ_{\pm} π / 2)}{\sqrt{r}}$ , where $d$ is the distance between the two slits. When the distance $d$ satisfies $k_{s p} d = π$ , the SPPs field can be simplified and expressed as

E_{s p}^{σ_{\pm}} = - \exp (i σ_{\mp} 2 α) \frac{\exp (i k_{s p} r)}{\sqrt{r}} .

From Eq. (2), it can be concluded that the amplitude of the SPPs generated by the slit pair is independent of the orientation angle and the phase changes linearly with $α$ . In Figs. 1(e) and 1(f), the simulated amplitude and phase distributions basically coincide with analytical results. The deviation between the simulated and analytical amplitude distributions could result from the scattering and the attenuation of SPPs during propagation. In the simulations, the nanoslits are etched on the 150 nm thick Au film and the substrate is glass. The wavelength of the circularly polarized incident light is 632.8 nm and the corresponding wavelength of excited SPPs $λ_{s p}$ is 606 nm. The width and length of the nanoslit are 50 nm and 150 nm, respectively, and the distance $d$ between the two orthogonal slits is 303 nm.

For the traditional semicircular slit plasmonic lens illuminated by circularly polarized light, the generated SPPs can be expressed as $E_{s p}^{σ_{\pm}} = \exp (i σ_{\mp} θ)$ , implying a uniform amplitude and a spiral phase distribution along the semicircular slit. For LCP light, the spiral phase which leads to the transverse shift of the SPPs focus varies from 0 to $π$ clockwise, as schematically given in Fig. 2(a). Due to the limited intrinsic spiral phase, the separation $s$ between the two SPPs focuses generated by LCP and RCP light has been restricted to $λ_{s p} / π$ [29–31,34]. Utilizing circularly polarized vortex incident beam, an extra spiral phase is imprinted on the generated SPPs and the field distribution can be represented as $E_{s p}^{σ_{\pm}} = \exp (i σ_{\mp} θ + l θ)$ , where $l$ is the topological charge of the vortex beam [33]. Thus, the deviation of SPPs focus from the center can be effectively enlarged, whereas the displacement of the SPPs focuses generated by LCP and RCP light changes synchronously. Thus, the separation $s$ between the two focuses remains to be unvaried. That’s because the extra spiral phase $l θ$ resulting from the OAM is independent from the spiral phase $σ_{\mp} θ$ induced by the SAM.

Fig. 2 The spiral phase for SPPs generated by the semicircular slit plasmonic lens (a). The schematic diagram of the designed semicircular SPPs lens consisted of orthogonal slit pairs (c) and the corresponding total spiral phase including intrinsic spiral phase $σ_{\mp} θ$ and additional spiral phase $σ_{\mp} 2 α$ (b).

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Based on the above analyses, enlarging the separation between the two SPPs focuses needs to increase the SAM induced spiral phase. Considering the SPPs field described by Eq. (2), arranging orthogonal slit pairs along a semicircle and changing the orientation angle $α$ with respect to the radial direction should be able to introduce another SAM related spiral phase, as schematically shown in Fig. 2(b). Thus, the total phase of SPPs is the combination of intrinsic spiral phase $σ_{\mp} θ$ and the additional spiral phase $σ_{\mp} 2 α$ caused by the rotation of slit pair. Omitting the constant term, the SPPs filed generated by the semicircular plasmonic lens shown in Fig. 2(c) can be described by:

E_{s p}^{σ_{\pm}} = \exp [i σ_{\mp} (θ + 2 α)] .

Based on Eq. (3), we can expect that by changing the orientation angle of the slit pairs, a larger separation

s

for the two SPPs focuses generated by LCP and RCP light can be achieved. There exist geometric and dynamic phase for SPPs modulation [24,36,39]. Resulting from the interaction between the subwavelength structures and circularly polarized light, the geometric phase is spin-dependent phase [27,29,30,36,39]. The dynamic phase relies on the radial position change of the structure, which makes it spin-independent [24,39]. For the semicircular SPPs lens in Fig. 2(b), the radial positions of the orthogonal slit pairs are all the same and both the intrinsic spiral phase

σ_{\mp} θ

and the additional spiral phase

σ_{\mp} 2 α

are spin-dependent. Therefore, both of them are geometric phase.

To generate the required spiral phase, the orientation angle $α$ can satisfy $α = m θ / 2$ , where m is an arbitrary constant determining the additional spiral phase. Thus, the total spiral phase of the excited SPPs is

Φ_{s p}^{σ_{\pm}} = σ_{\mp} (1 + m) θ .

To enhance the intensity of the SPPs, three columns of slit pairs with a period of

λ_{s p}

are adopted and the inner radius is 5 μm. For semicircular plasmonic lenses with m = 0, 1, 2, 3, 4, 5, the simulated SPPs intensity distributions for LCP and RCP light are shown in Figs. 3(a) and 3(b), respectively. White dashed lines are drawn along the center as the visual reference to show the transverse displacement. For LCP and RCP incident light, the SPPs focuses gradually deviate from the center downward and upward with the growth of m, respectively. That’s because the range of the total spiral phase gradually increases from 0~

π

to 0~

6 π

with a step of

π

for m = 0, 1, 2, 3, 4, 5, according to Eq. (4). In the 3D free space, the physical mechanism of SHEL can be classified into two kinds of geometric phase: the Rytov-Vladimirskii-Berry (RVB) phase and the PB phase [13,40–42]. The RVB gradient phase in the momentum space results in spatial shift (spin Hall shift in real space), while the PB gradient phase in the real space leads to angular shift (spin Hall shift in the k space) [40,41]. The spin-dependent transverse shift of the SPPs focus is attributed to the spiral phase which is composed of PB phase generated by nanoslits with spatial variant orientation angles [36]. Thus, the spiral phase can be regarded as a PB gradient phase along the semicircle in the real space. The spiral phase leads to the shift of the SPPs wavevector

k_{s p}

from the radial direction and further gives rise to the spin-dependent focusing. Hence, the spin-dependent SPPs focusing here should belong to the angular shift category.

Fig. 3 Simulated SPPs intensity distributions for the m = 0, 1, 2, 3, 4, 5 semicircular plasmonic lenses illuminated by LCP (a) and RCP (b) light. The SPPs focuses deviate gradually from the center with the increase of m.

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The normalized transversal distributions of the SPPs focuses are given in Figs. 4(a) and 4(b) to show the transversal displacements quantitatively. The full width half maximums (FWHMs) of the SPPs focuses are hardly affected by the change of m and remain to be around 230 nm. The exact displacements of the SPPs focuses are also extracted and shown in Figs. 4(c) and 4(d). An obvious linear relation between the displacement of the focuses d and m (as well as the spiral phase) is found and can be fitted as:

d_{m} = σ_{\mp} (m + 1) d_{0} = \frac{Φ_{s p}^{σ_{\pm}}}{π} d_{0} .

_{$d_{0}$}≈128 nm represents the transverse shift of the SPPs focus generated by the m = 0 SPPs lens which is equivalent to the semicircular slit lens [33]. Theoretically estimated displacements of the focal spots can be obtained with

d_{m} = σ_{\mp} χ_{m + 1} / k_{s p}

, where

χ_{m + 1}

is the first non-null zero point of the deviation of the Bessel function

J_{m + 1} (y)

[33]. Huygens-Fresnel principle for SPPs [33,34,43] is also adopted to analytical study the transverse displacement of the SPPs focus. The SPPs filed is the interference of SPPs emitted by the secondary sources (the orthogonal slit pairs) along the semicircular lens. Analytical and theoretical displacements agree well with the simulated and fitted results, which prove the feasibility of the proposed mechanism. According to Eq. (5), with the increase of m and the spiral phase, the transversal displacement of SPPs focus can be enlarged. For the m = 5 semicircular plasmonic lens, the transverse shift of the focus is

d_{5} = 6 d_{0}

, which is six times larger than the displacement acquired with the traditional semicircular slit plasmonic lens. And the separation between the two SPPs focuses generated by LCP and RCP light reaches

s = 2 d_{5} = 12 d_{0}

(≈1.5 μm), which is large enough to effectively reduce the crosstalk between the two SPPs focuses in applications like on-chip communications. To experimentally fabricate the proposed structure, the Au film with a thickness of 150 nm can be evaporated on the glass substrate by using e-beam evaporator [27,35,44]. Then, the orthogonal nanoslits can be patterned by using the focused ion beam (FIB) machine [21,27,30,35]. For the experimental configuration, the desired circularly polarized light with a wavelength of 632.8 nm can be generated by using a polarizer and quarter wave plate and should illuminate the sample from the backside. The distributions of the focusing SPPs excited by the semicircular lenses can be measured by near-field scanning optical microscope (NSOM) technique [21,30,33,35,36].

Fig. 4 Transverse profiles of the SPPs focus generated by LCP (a) and RCP (b) light. (c) and (d) show the linear relation between the displacement of the SPPs focuses and m.

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Moreover, the proposed approach can provide extra flexibility to control the SPPs focus, considering that the constant m can be both negative and positive. With an m = −4 semicircular plasmonic lens, the direction of the spiral phase is reversed and increases counterclockwise (clockwise) for LCP (RCP) light. Thus, compared with the m = 2 lens, the displacement of the SPPs focuses generated by LCP and RCP light will be reversed, but the distance of the transverse shift is the same. The intensity of the SPPs focus is given in Fig. 5(a). Special attention should be paid when m = −1, the spiral phases for both LCP and RCP incident light disappear according to Eq. (4) and the transverse displacements of the SPPs focuses are zero according to Eq. (5). As shown in Fig. 5(b), the SPPs focuses are both in the center, which means the SPPs lens is spin-independent in this case. Furthermore, the constant m can be not only integer but also decimal, which means the displacement of SPPs focus can change continuously. Arbitrary desired displacement of the SPPs focus can be obtained by calculating the m value with Eq. (5). As shown in Fig. 5(c), the simulated displacement of the SPPs focus generated by m = 2.5 plasmonic lens is consistent with the analytical value (≈440 nm). Furthermore, inheriting from the traditional semicircular slit plasmonic lens [33], the displacement of the SPPs focus is immune to the change of the radius of the lens. Figure 5(d) gives the SPPs distributions generated by an m = 2, r = 2.5 μm semicircular lens, the displacement of the SPPs focus is almost the same as the m = 2, r = 5 μm lens (≈390 nm). In Refs [22,44], spin-dependent SPPs focusing is achieved with vertically arranged slits and amplitude modulation [22] or combined phase modulation [44] is adopted, which makes the size of the SPPs lens dozens of micrometers. With semicircular SPPs lens here, the size can be reduced to only 5 μm in Fig. 5(d) because the spin-dependent SPPs focusing is independent of the radius. Moreover, the pure spiral phase modulation is more simple and flexible, which can enlarge, reverse or eliminate the spin-independent SPPs focusing.

Fig. 5 (a) For negative an m = −4 lens, the displacements of SPPs focuses generated by LCP and RCP light will be reversed. (b) The spiral phase disappears and the SPPs focusing is spin-independent when m = −1. And m can also be decimal (c), which means the displacement of the SPPs can change continuously. (d) is the SPPs focuses generated by a r = 2.5 μm semicircular lens.

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3. Conclusions

In conclusion, we have demonstrated the significant enlargement of the spin-dependent transverse displacement of SPPs focus with a semicircular plasmonic lens consisted of orthogonal slit pairs. Introducing additional spiral phase by changing the orientation angle of the slit pairs plays an essential role in modulating the SPPs focus. For incident light with a wavelength of 632.8 nm, the splitting of the SPPs focuses generated by LCP and RCP light can readily reach 1500 nm, which can effectively avoid the crosstalk between the SPPs focuses in applications like on-chip communications. Moreover, the theoretically obtained relationship between the displacement and the total spiral phase suggest that both the direction and magnitude of the displacement can be controlled with the proposed approach.

Funding

National Natural Science Foundation of China (NSFC) (11704231, 11804199); China Postdoctoral Science Foundation (2017M622252).

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Enlarging spin-dependent transverse displacement of surface plasmon polaritons focus

Abstract

1. Introduction

2. Results and discussions

3. Conclusions

Funding

References

Cited By

Figures (5)

Equations (5)

Optics Express