Photonic spin Hall effect: a new window in D-shaped fiber by weak measurements

Famei Wang; Zhijie Sun; Chunhong Xu; Lin Yang; Chao Liu; Tao Sun; Paul K. Chu

doi:10.1364/OE.27.014064

1. Introduction

The photonic spin Hall effect (SHE) is the phenomenon that the components of the left and right-handed circularly polarized light are transversely split into two beams along the direction perpendicular to the incident plane according to their rotational directions when a linearly polarized light passes through an inhomogeneous medium [1–3]. The mechanism behind photonic SHE is due to spin-orbit coupling and the refractive index gradient. The photonic SHE is a sensitive physical effect as the transverse shift is very sensitive to the change in structural parameters in an inhomogeneous medium and therefore, it has tremendous potential as a precise measurement technique for monitoring naometal film thickness [4], graphene layer numbers [5], axion angle of topological material [6], and the distance of Weyl nodes in the momentum space [7]. Moreover, it also opens up a possible way to control the spin state of photons and develops the next generation of photonic spin Hall devices [8,9].

Spurred by advances in micro-fabrication technology, many methods have been developed to enhance the photonic SHE which has consequently attracted much interest for the unique characteristics and practical potential. The phenomenon was discovered in 2004 when Onoda et al. investigated the ray of light in a photonic crystal without the inversion symmetry in which the Hall effect can be magnified [10]. Afterwards, Luo et al. proposed an enhanced and switchable SHE of light near the Brewster angle on reflection [11]. The spin-dependent splitting of 3.2μm was achieved in the study. In 2016, the photonic SHE based on the surface plasmon resonance (SPR) effect was theoretically investigated by Zhou et al. and Tan et al. accomplished enhanced photonic SHE by making use of long-range SPR (LRSPR) [12]. The maximum spin-dependent splitting reached 7.85 μm for an incident Gaussian beam of 632.8 nm [13]. Zhang et al. demonstrated that the spin-dependent displacement produced by the photonic SHE depended on the optical axis direction, doping concentration, and interband transitions [14]. In addition, the photonic SHE in transmission has also been widely explored in inhomogeneous anisotropic media [15], uniaxial crystal plate [16], polymer [17], metamaterials [18], and spherical geometry [19].

So far, most researchers have focused on the photonic SHE by coating the prism surface but systematic investigations of the photonic SHE in surface coatings based on fiber have rarely been reported. Although the performance of the prism coated with multilayer films is quite good, its structural configuration is bulky because there are many required optical and mechanical components thus posing challenges for remote sensing. In recent years, the optical fiber based SPR sensing technique is considered a possible route to miniaturization [20–22]. D-shaped optical fibers have been shown to be a good substitute for prisms due to the smaller footprint, easier system integration, and cost effectiveness. They have been widely used in a wide range of sensing applications, including sensing changes in refractive index, temperature, magnetic field, pressure, biochemical properties, and so on [23]. Since the layer thickness on the fiber is generally on the order of nanometers, variations of most physical properties, effects, and parameters tend to be quite small for such a small layer thickness and can be regarded as weak effects. Since its scale has reached the scope of the micro system, many quantum effects emerge and even play an important role under some conditions. This is challenge to traditional measurements.

Here, we theoretically investigate the photonic SHE based on the D-shaped fiber coated by Ag-Ni alloy/Si layers during SPR excitation. We analyze the effects of the Ag-Ni alloy layer and silicon layer to find the optimal condition. A maximum transverse shift of 420μm several times larger than previous results is observed. The large transverse shift and angular sensitivity derived from the new design is applicable to photonic SHE in nanophotonic devices.

2. Model and method

Figure 1(a) displays the experiment setup. A Gauss beam generated by a He-Ne laser firstly passes through a short focal length lens (L1) to produce an initially linearly polarized focused beam. And then, the light beam is input into the D-shaped fiber of multilayered structure through a short focal length lens (L2) to collimate the beam and make the beam shifts insensitive to the distance between L2 and the optical spectrum analyzer (OSA), so that we can measure the amplified beam shifts accurately [4,20]. Finally, a personal computer (PC) is used to analyze spectral data collected by the OSA to obtain the transverse shift of the photonic SHE. To enhance the transverse displacement of the horizontal polarization state, we propose a multilayer structure (seen in Fig. 1) composed of the optical fiber, Ag-Ni alloy, silicon, and air. In this structure, we select the following parameters: $n_{1} = 1.515$ , $n_{2} = 0.4609 + 1.7356 i$ , $n_{3} = 3.92 + 0.01 i$ , $n_{4} = 1.0$ for the incident wavelength of 632.8 nm, and $d_{2}$ and $d_{3}$ represent the thicknesses of the alloy layer and silicon layer, respectively. The refractive index of the Ag-Ni alloy is determined based on the available literature [24]. If the proper incident angle and thicknesses of the Ag-Ni alloy layer and silicon layer are chosen, the photonics SHE will be excited as long as the light beam is reflected at the fiber-alloy-Si interface. $x y z$ is a laboratory coordinate system and the $z$ axis is perpendicular to the interface. ( $x_{i}$ , $y_{i}$ , $z_{i}$ ) and ( $x_{r}$ , $y_{r}$ , $z_{r}$ ) represent the coordinate systems of the incident and reflected light, respectively. The light beam impinges the multilayer structure from the fiber core at an incident angle $θ$ along the $z_{i}$ direction. When $z = 0$ (interface between the fiber and alloy layer) is reflected along the $z_{r}$ direction, splitting of left-and right-handed circularly polarized components occurs due to the photonic SHE. We consider an incident Gaussian beam with an angular spectrum of [25]:

{\tilde{E}}_{i} (k_{i x}, k_{i y}) = \frac{w_{0}}{\sqrt{2 π}} \exp [- \frac{w_{0}^{2} (k_{i x}^{2} + k_{i y}^{2})}{4}],

where

w_{0}

is the waist of the Gaussian beam. According to the continuity condition of the electric field boundary, the relationship between the angular spectrum of the reflected field and angular spectrum of the incident field is described as follows [26]:

[\begin{matrix} {\tilde{E}}_{r}^{H} \\ {\tilde{E}}_{r}^{V} \end{matrix}] = [\begin{matrix} r_{p} & \frac{k_{r y} (r_{p} + r_{s}) \cot θ}{k_{0}} \\ - \frac{k_{r y} (r_{p} + r_{s}) \cot θ}{k_{0}} & r_{s} \end{matrix}] \times [\begin{matrix} {\tilde{E}}_{i}^{H} \\ {\tilde{E}}_{i}^{V} \end{matrix}],

where

{\tilde{E}}_{r}^{H, V}

stands for the circular components of the reflection field,

H

and

V

denote horizontal and vertical polarization, respectively,

k_{0} = 2 π / λ

is the vector of the wave in vacuum,

λ

is the wavelength of the incident light, and

r_{p}

and

r_{s}

represent the Fresnel reflection coefficients for

H

and

V

polarization states, respectively. According to Eqs. (1) and (2), we can first get the expressions of the reflected angular spectrum [4]:

{\tilde{E}}_{r}^{H} = \frac{r_{p}}{\sqrt{2}} [\exp (+ i k_{r y} δ_{r}^{H}) {\tilde{E}}_{r +} + \exp (- i k_{r y} δ_{r}^{H}) {\tilde{E}}_{r -}] and

{\tilde{E}}_{r}^{V} = \frac{i r_{s}}{\sqrt{2}} [- \exp (+ i k_{r y} δ_{r}^{V}) {\tilde{E}}_{r +} + \exp (- i k_{r y} δ_{r}^{V}) {\tilde{E}}_{r -}] .

Here

δ_{r}^{H} = (1 + r_{s} / r_{p}) \cot θ / k_{0}

,

δ_{r}^{V} = (1 + r_{p} / r_{s}) \cot θ / k_{0}

. Based on Eqs. (3) and (4), the complex amplitude for the reflected fields can be obtained as follows:

E_{r \pm} (H, V) = \int d k_{r x} d k_{y} {\tilde{E}}_{r \pm} (H, V) \exp [i (k_{r x} x_{r} + k_{r y} y_{r} + k_{r z} z_{r})] .

For the GWSPR configuration, the reflection coefficients can be written as

r_{p, s} = \frac{r_{12} + r_{234} \exp (i ϕ_{2})}{1 + r_{12} r_{234} \exp (i ϕ_{2})} and

ϕ_{2} = \frac{4 π}{λ} n_{2} d_{2} \cos θ_{2},

where

r_{234}

is the reflection coefficient at the interface of layers

2

,

3

,and

4

obtained from [27]:

r_{234} = \frac{r_{23} + r_{34} \exp (i ϕ_{3})}{1 + r_{23} r_{34} \exp (i ϕ_{3})} and

ϕ_{3} = \frac{4 π}{λ} n_{3} d_{3} \cos θ_{3},

where

r_{12}

,

r_{23}

, and

r_{34}

are the reflection coefficients at the single interfaces as follows [28]:

r_{m n} = \frac{\sqrt{ε_{m}} \cos θ_{m} - \sqrt{ε_{n} - ε_{m} \sin^{2} θ_{m}}}{\sqrt{ε_{m}} \cos θ_{m} + \sqrt{ε_{n} - ε_{m} \sin^{2} θ_{m}}}

for s-polarization and

r_{n m} = \frac{ε_{n} \cos θ_{m} - \sqrt{ε_{m}} \sqrt{ε_{n} - ε_{m} \sin^{2} θ_{m}}}{ε_{n} \cos θ_{m} + \sqrt{ε_{m}} \sqrt{ε_{n} - ε_{m} \sin^{2} θ_{m}}}

for p-polarization, where

m, n = 1, 2, 3, 4

, respectively. According to Eqs. (6)–(11), we establish the general beam-propagation model to describe the light beam reflected from the four-layer SPR model. When the transmission distance

z_{r}

of a beam is constant, the shifts in the light beam centroid can be expressed as follows [27,28]:

Fig. 1 (a) Experimental setup, (b) and (c) detailed schematic of the photonic SHE in the multilayer structure composed of the optical fiber with $Δ y_{-}$ and $Δ y_{+}$ denoting the transverse displacement of the left and right circular polarization components, respectively.

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Δ y_{\pm} = \frac{\iint y_{r} {| E_{r \pm} (H, V) |}^{2} d x_{r} d y_{r}}{\iint {| E_{r \pm} (H, V) |}^{2} d x_{r} d y_{r}} .

Based on Eqs. (5)–(11), the transverse spin-dependent displacements of the two spin components can be defined as follows [4]:

Δ y_{\pm}^{H} = \mp \frac{λ}{2 π} [1 + \frac{| r_{s} |}{| r_{p} |} \cos (ϕ_{s} - ϕ_{p})] \cot θ and

Δ y_{\pm}^{V} = \mp \frac{λ}{2 π} [1 + \frac{| r_{p} |}{| r_{s} |} \cos (ϕ_{p} - ϕ_{s})] \cot θ,

where

r_{p, s} = | r_{p, s} | \exp (i ϕ_{p, s})

,

ϕ_{p, s}

are the phase angles of the Fresnel reflection coefficient.

3. Results and discussion

Figure 2 presents optical field distributions of the photonic SHE based on the D-shaped fiber for different incident angles by using commercial software (COMSOL multiphysics). As evidenced by the fact that the optical field energy in the core gradually transfers to Ag-Ni/Si layers along the left and right horizontal directions with increasing the incident angles, it can be confirmed that the transformation of propagating modes occurs in the photonic SHE based on the D-shaped fiber.

Fig. 2 Optical field distributions of the D-shaped fiber of multilayer structure for different incident angles.

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To study the influence of different metal layers on the enhanced photonic SHE under the condition of SPR, we first investigate the relationship of the Au, Ag, alloy layer with the SPR. As shown in Fig. 3(a), the Fresnel reflectance changes with the incident angle when the metal layer is Au, Ag, alloy, respectively. The Fresnel reflection coefficient shows a narrow resonant dip near the resonant angle due to excitation of the SPR. Figure 3(a) also shows that the Fresnel reflection coefficient of the alloy at the resonance angle is remarkably less than that of the pure metal. According to Eqs. (13) and (14), the transverse displacement of the reflected light for the H-polarization state is proportional to the ratio of $| r_{s} / r_{p} |$ . That is, the large ratio $| r_{s} / r_{p} |$ near the resonant dip enhances the transverse shifts of H-polarization [24]. Figure 3(b) shows that the ratio of $| r_{s} / r_{p} |$ has a greater value at the SPR resonance angle for the horizontal polarized light in the Ag-Ni alloy structure. Figure 3(c) shows the maximum ratio of $| r_{s} / r_{p} |$ and corresponding resonant angle for different thicknesses of the alloy layer. The resonant angle varies from 51.2° to 52.4° with increasing $d_{2}$ when $d_{3} = 0$ nm and the optimal thickness of the alloy layer is 50 nm for the resonant angle at 51.5° (Fig. 3(d)). Therefore, the transverse displacement of H-polarization state depends very much on the ratio of $| r_{s} / r_{p} |$ , which will be explained in details below.

Fig. 3 (a) Dependence of the Fresnel reflection coefficients shifts on the metal layer and incident angle for $d_{3} = 0$ ; (b) Value of $| r_{s} / r_{p} |$ changing with the incident angles for $d_{3} = 0$ ; (c) and (d) Relationship of the value of $| r_{s} / r_{p} |$ and incident angle for different thicknesses of the alloy layer.

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Compared with the single alloy, the method of placing an extra silicon layer on the alloy layer provides larger absorption and hence, the refractive index variation gradient enhances the photonic SHE [25]. Figure 4 presents the multilayer fiber structure consisting of a layer of silicon on the alloy layer. The silicon layer has a major impact on the ratio $| r_{s} / r_{p} |$ as shown in Fig. 4. The resonant angle moves to a larger angle and the full-width-at-half-maxima (FWHM) of reflectance broadens [22]. To achieve the maximum ratio of $| r_{s} / r_{p} |$ , the thicknesses of the silicon layer and alloy layer are 1 nm and 49 nm, respectively.

Fig. 4 Dependences of the ratio $| r_{s} / r_{p} |$ on the incident angle for different thicknesses of the alloy and silicon layer.

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To optimize the parameters to achieve the maximum transverse shifts, the contour maps of the transverse shifts are plotted in Fig. 5 for the H-polarization state and V-polarization state, respectively. The thicknesses of the alloy layer and silicon layer are in the range of 48 ~50 nm and 0 ~2 nm, respectively. Figure 5(a) displays the variation of the transverse displacements with the alloy layer and silicon layer thicknesses and there are optimal thicknesses of the alloy layer and the silicon layer due to excitation of the SPR. However, with respect to the V-polarization state, SPR cannot be excited so that there are no enhanced beam shifts [Fig. 5(b)]. Therefore, to analyze the enhancement of the photonics SHE, we choose the H-polarization state to analyze the reflection coefficient. Figure 5(a) shows that the maximum transverse displacement reaches 496 μm and the optimal alloy layer and silicon layer thicknesses are 48.86 nm and 1 nm, respectively. However, it is noted that the requirement for the thickness accuracy of the metal layer is too high and it is difficult to achieve by current processing technologies [29–31]. Therefore, to analyze the enhancement of the photonics SHE, we select 49 nm and 1 nm as the optimal thicknesses of the alloy layer and the silicon layer.

Fig. 5 Dependence of the transverse beam shifts on the alloy layer and silicon layer thicknesses: (a) H-polarization state and (b) V-polarization state.

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In Fig. 6 we show the dependence of the transverse spin-dependent displacements on the incident angle for the H-polarization state. The black and red curves represent the transverse displacement of the right and left circular polarization components, respectively. Figure 6 shows that the transverse spin-dependent displacements vary with the incident angle and there are optimal values for both the transverse spin-dependent displacements and incident angle due to excitation of the SPR. At the resonance angle of 53.2°, the maximum transverse displacement is 420 μm, which is 36 times larger than those reported previously for refraction [28]. In general, the transverse beam shift of photonic SHE is not only remarkably dependent on the beam waist of Gaussian beam, but also on the ratio of $| r_{s} / r_{p} |$ and the phase angles of the Fresnel reflection coefficient. In this work, the reasons for the giant transverse shift are dominated by the ratio of $| r_{s} / r_{p} |$ and the phase angles, rather than beam waist based on Eqs. (13) and (14). The inherent physics behind this giant transverse displacement is attributed to the fact that Ag-Ni/Si heterostructure provides a larger light absorption and hence the refractive index variation gradient, which can effectively enhance the photonic SHE due to SPR [12,25].

Fig. 6 Dependence of the transverse beam shifts on the incident angles for $d_{2} = 49 n m$ and $d_{3} = 1 n m$ .

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Figure 7 illustrates the dependence of the transverse shift $Δ y_{+}$ as a function of the refractive index of the external environment and incident angle. Figure 7 presents the influence of the spin-dependent splitting on photonic SHE changes with the refractive index of the external environment from 1.0 to 1.02. The optimal resonant angle shows red shift and the peak of the transverse shift $Δ y_{+}$ decreases when the refractive index of the external environment rises from 1.0 to 1.02. The angular sensitivity ( $S_{θ} = Δ θ / Δ n$ ) for different refractive indexes of the external environment can be calculated from Fig. 7. The maximum variation in the resonance angle is $Δ θ = {1.146}^{°}$ when the refractive index and maximum angular sensitivity is as high as 114.6°/RIU. Our results suggest a new means to study the angular sensitivity of gas sensors [32,33].

Fig. 7 Spin-dependent splitting in photonic SHE changing with the refractive index of the external environment ( $d_{2} = 48.86 n m$ and $d_{3} = 1 n m$ ).

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4. Conclusions

We investigate the enhanced photonic SHE based on the D-shaped fiber under the condition of the SPR effect. The maximum transverse shift of the H-polarization state of 420 μm is achieved by optimizing the parameters such as the thicknesses of the Ag-Ni alloy layer and silicon layer. The transverse beam displacement reported in this study is much greater than previously reported results measured for the prism-metal interface. This study of photonic SHE based on the D-shaped fiber is expected to spur the development of quantum weak measurement theory for optical fibers as well as concise, cost-effective, and precise measurement methods to study optical fiber coatings.

Funding

National Natural Science Foundation of China (NSFC) (51474069); China Postdoctoral Science Foundation funded project (2016M591510); Natural Science Foundation of Heilongjiang Province (E2016007); City University of Hong Kong Strategic Research Grant (SRG) (7005105); Major project of Applied Technology Research and Development Plan of Heilongjiang Province (GA18C001).

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Photonic spin Hall effect: a new window in D-shaped fiber by weak measurements

Abstract

1. Introduction

2. Model and method

3. Results and discussion

4. Conclusions

Funding

References

Cited By

Figures (7)

Equations (14)

Optics Express