1. Introduction

Spin Hall effect of light (SHEL) is a peculiar phenomenon, which means that when a light beam is reflected or transmitted from the interface, due to the spin-orbit angular momentum coupling, there will be the transverse shift (TS) of the left-handed circularly polarized and the right-handed circularly polarized light perpendicular to the incident plane [1,2]. In 2008, Hosten and Kwiat first measured the spin splitting of the transmitted Gaussian beam through an air-glass interface [3]. Then, a great deal of studies focused on the enhancement and regulation of SHEL [4–6]. Most recently, Zhu et al. proposed a giant spatial spin separation which exceeds the upper limit with high energy efficiency based on a wave-vector-varying PB phase [7]. Luo et al. and Kong et al. proved theoretically and experimentally at air-glass interface that there was a large TS near the Brewste’s angle, but zero at the Brewster’s angle, respectively [8,9]. However, the TS is only a fraction of wavelength generally. Recently, optical differential operation and image edge detection have attracted great interest because their promising application prospect in image processing [10–14]. Now, the SHEL has been widely studied in many new materials, such as epsilon-near-zero metamaterials [15–17], chiral metamaterials [18], topological insulators [19], two-dimensional atomic crystals [20], graphene materials [21], PT symmetric metamaterials [22] and so on. Thanks to the maturity of weak value amplification (WVA) technology [23,24], it has made great applications in precision measurement [25,26], such as identifying the graphene layer numbers [27,28], precise measurement of optical conductivity of atomically thin crystal [29], detection of chemical reaction rate [30], and measurement of ion concentration [31]. In addition to the precise measurement reports of the above physical quantities. Due to the SHEL originates from the gradient of refractive index (RI), it is extremely sensitive to the change of RI, and the RI sensor is also a promising application [32–34].

On the other hand, the Kretschmann configuration has been widely utilized to achieve giant TS owing to the excitation of surface-plasmon resonance (SPR) [35,36]. Such resonance makes electrons absorb the energy of light, and the reflected light is greatly weakened. At this time the incident light has the same frequency and wave vector as surface plasmons (SPs) at a specific angle, which is called resonance angle ($\theta _{SPR}$) [37]. A series of studies showed that the TS can be enhanced near the $\theta _{SPR}$ [38,39]. Zhu et al. discussed that more interesting phenomena will occur near the resonance angle when a Laguerre-Gaussian light is incident [40]. Tan et al. enhanced the SHEL by taking advantage of long-range SPR (LRSPR) [41]. Zhou et al. realized SPR refractive index sensor through WVA [42]. However, the above SHEL about SPR researches are based on the traditional noble metals. While sodium metal has lower cost and intrinsic loss [43], due to its low intraband damping rate, it always been regarded as an ideal SPR material. Quite recently, Wang et al. fabricated stable high-performance sodium-based plasmonic devices in the near-infrared based on a thermo-assisted spin coating process [44]. They proved that sodium has lower loss and higher figure of merit than traditional metals in the near in-frared band. Therefore, sodium offers a new opportunity to realize high performance SPR sensing beyond noble metals. Combine with the superiority of sodium and great potential application in precision measurement and sensing. To the best of our knowledge, sodium-based SPR of SHEL sensing is rarely studied before.

In this paper, we theoretically establish a controllable RI sensing and multi-functional detecting multilayered structure containing sodium. Firstly, the effects of sodium on TS and polymethyl methacrylate (PMMA) layer on resonance angle are discussed. The maximum value of traverse shift ($TS_{max}$) can be obtained with appropriate parameters. Moreover, we compare the traditional noble metals with sodium and find that sodium has obvious advantages in TS and sensitivity. Next, the quantitative relationship between TS and RI of the sensing medium is established. The measurement range can be controlled by Fermi energy, PMMA and sodium. Specially, the sensitivity can be controlled by sodium. Compared with other SPR multilayered structures of SHEL sensing, our structure has the advantages of wide range RI sensing and adjustable sensitivity. Then, the $TS_{max}$ are quite different in the range of $E_{\mathit {f} } <0.517$ eV and $E_{\mathit {f} } > 0.517$ eV respectivly. The resonance angle of this structure is concentrated around $71.1^{\circ }-71.4^{\circ }$, so this phenomenon can be used as an accurate measurement of resonance angle. Finally, starting from $71.11^{\circ }$, the Fermi energy can be measured in a wide range by increasing the incident angle every $0.04^{\circ }$.

2. Theory and models

As shown in Fig. 1, our structure is composed of BK7 glass, sodium metal film, PMMA layer, monolayer graphene sheet and sensing medium. The symbols of $n_{1}, n_{2}, n_{3}, n_{4}$ represent the RI of prism, sodium metal, PMMA layer and sensing medium, and $d_{2}$ and $d_{3}$ represent the thicknesses of metal and PMMA layer, respectively. Here $n_{1}=1.5049$, $n_{2}=\sqrt {\varepsilon _{2}}$, $n_{3}=1.4739$, $n_{4}=1.32$. The prism and sodium metal form a Kretschmann configuration. When a TM-polarized light beam is incident on the prism-metal interface, SPs can be excited. In the near-infrared band, sodium metal has its advanced performance and larger figure of merit. According to the Drude–Lorentz model [45,46], the dielectric function of sodium is

 

Fig. 1. Schematic of controllable refractive index sensing and multi-functional detecting structure (composed of BK7 glass, sodium film, PMMA layer, graphene sheet and sensing medium) by considering the SPR effect.

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In order to study the relationship between spin-dependent spatial splitting and external variables, we first establish a general transmission model of light reflection by using angular spectrum theory. The monochromatic polarized Gaussian light beam impinges upon the glass-metal interface with the incidence angle $\theta$, and incident wavelength is $1.2$ $\mu m$. The angular spectrum of Gaussian beam can be expressed as

Here, $\xi =k_{r y}\left (r_{p}+r_{s}\right ) \cot \theta _{i} / k_{0}$, $k_{ry}=k_{iy}$, $k_{0}$ is the wave number in free space. $r_{p}$ and $r_{s}$ denote the Fresnel reflection coefficients for $H-$ and $V-$ polarization states.

The calculation of the reflected beam TS requires the explicit solution of the boundary conditions. Thus, the detailed expression of Fresnel coefficients of this model can be obtained by the transmission matrix method [48]:

According to Eqs. (3)–(6), after the inverse Fourier transform, two circular components of the reflected fields can be obtained

Based on Eqs. (7)–(9), TS at $z_{r}=0$ can be derived as follows:

Equations (10) and (11) are the theoretical formulas for detecting the TS of the strcture. Here we only discuss the TS of p-polarized incidence.

3. Result and discussion

Because the SPR can affect the reflection coefficient of metal medium, it has a great influence on the TS. Here Fig. 2 reveals the variation of the TS with the incident angle under different thicknesses of sodium and PMMA layers. It is found that the thickness of PMMA has little effect on the TS, but the resonance angle, where the point TS=0, increases with the thickness of PMMA layer. However, the resonance angle does not move when the thickness of sodium layer is changed. On the contrary, it has a dramatic change in the TS. The TS is smaller when the sodium thickness is not large enough. As the thickness of sodium layer reaches $100$ nm, the TS reaches $325$ nm. This is because when the thickness of the metal is small, there are almost no SPs. As the thickness of the metal increases, the SPs are excited by the evanescent wave which diffuses in the interface, resulting in a larger TS. But the penetration depth of evanescent wave in sodium film is limited. If the thickness of sodium is further increased, the TS will be attenuated.

 

Fig. 2. Relationship of TS changing with the incident angle ($\theta$) under different PMMA thicknesses ($d_{3}$) at $E_{\mathit {f} } = 0.517$ eV, in which red dashed line $d_{3}=60$ nm, green dotted line $d_{3}=80$ nm and blue solid line $d_{3}=100$ nm. (a) $d_{2}=60$ nm, (b) $d_{2}=80$ nm, (c) $d_{2}=100$ nm. $n_{4}=1.32$.

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An interesting phenomenon is found in the discussion of Fig. 2. When two thicknesses reach the appropriate parameters, the TS evolves from asymmetric to symmetric distribution. We compare the conventional noble metal gold [50] and silver [51] with sodium in this structure, and find that the symmetrical distribution of $\delta _{-}$ can be obtained by slightly changing the thickness of metal and PMMA near the $TS_{max}$ point, as shown in Fig. 3(a). Importantly, in the symmetric distribution, the TS of sodium is superior. This is the huge value of $|r_{s}|/|r_{p}|$ result in a larger TS. At the same time, the phase mutation of $r_{p}$ appears at the resonance angle, resulting in the symmetrical distribution of TS. The longer tunneling depth of the evanescent wave and the larger range $r_{p}$ resonance peak, which is also the advantage of sodium.

 

Fig. 3. (a) Dependences of the symmetric TS on the incident angle $\theta$ for different metals at $n_{4}=1.32$. (b) The relationship between TS and RI of sensing medium by taking the respective resonance angle of three kinds of differrent metals as the incident angle. Red solid line is sodium ($d_{2}=101$nm, $d_{3}=98$ nm). Green solid line is gold ($d_{2}=48$ nm, $d_{3}=81.03$ nm). The blue dotted line is silver ($d_{2}=48$ nm, $d_{3}=76$ nm). $E_{\mathit {f} } = 0.517$ eV.

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We know from Fig. 3(a) that the resonance angles of sodium, gold and silver are $71.25^{\circ }$, $65.38^{\circ }$ and $65.28^{\circ }$, respectively. With respective resonance angle as incident angle, the function of TS with RI is drawn, as shown in Fig. 3(b). By observing the TS of the small RI change of three different metals from 1.3197-1.3200, we can know that the sensitivity $(S_{\delta } ={\triangle \delta }/{\triangle n_{4} })$ of sodium is the largest, which is 1844.9 $\mu$m/RIU, followed by silver, 1393.3 $\mu$m/RIU, and finally gold, 1274.9 $\mu$m/RIU. As previously analyzed, the larger TS of sodium leads to its greater sensitivity, which is also the external manifestation of the stronger SPR of sodium compared with other metals. The sensitivity analysis of sodium will be discussed in detail in Fig. 4. It can be seen from Fig. 3 that $\delta _{-}$ is symmetrical distribution. From Eqs. (10) and (11), it is known that the $\delta _{+}$ and $\delta _{-}$ are also symmetrical distribution each other. Therefore, we can control the left and right circularly polarized light appear alternately. This phenomenon also provides a great application prospect for controlling the characteristics of polarized light, such as polarization selectors, etc.

 

Fig. 4. (b) shows the change of TS with RI of sensing medium. TS as a function of RI of sensing medium and incident angle (a), Fermi energy (c). The right column describes the change of TS with the RI of sensing medium under different sodium layer thickness ($d_{2}$) at $E_{\mathit {f} }=0.517$ eV, in which red dashed line $d_{2}=60$ nm, green dotted line $d_{2}=80$ nm, blue solid line $d_{2}=100$ nm. (d) $d_{3}=60$ nm, (e) $d_{3}=80$ nm, and (f) $d_{3}=100$ nm.

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Because the SHEL originates from the gradient of the RI, it is very sensitive to the change of the RI of the sensing medium, which has the application prospect of the RI sensor. Firstly, we make $E_{\mathit {f} }= 0.517$ eV, $d_{2}= 93$ nm, $d_{3} =100$ nm. Figure 4(a) shows the variation of the TS with the incident angle $\theta$ and RI of sensing medium $n_{4}$. The $n_{4}$ is almost linear with $\theta$, so we can select different RI materials as the sensing medium, and the selection range is $1.1-1.4$. Here, $n_{4} = 1.32$ is selected as an example, and the corresponding resonance angle is $71.4^{\circ }$. Fixed $71.4^{\circ }$ as the incident angle. When the biomolecules enter the sensing medium, the RI (biomolecular concentration) of the sensing medium variation can be accurately measured by observing the change of TS, as shown in Fig. 4(b). The sensitivity $S_{\delta }$ is $206$ $\mu$m/RIU ranged 1.3195-1.3210, and the accuracy is two orders of magnitude lower than the original RI.

Then, the biomolecules continue to enter the sensing medium. We can change the detection range of RI by adjusting the Fermi energy of graphene, as shown in Fig. 4(c). What’s more, $E_{\mathit {f} }=\hbar v_{f} \sqrt {\pi n_{g}}$ in the graphene, where $n_{g}$ is the charge density, so we can control $E_{\mathit {f} }$ through adjusting $n_{g}$ controlled by an applied gate voltage continuously. We can observe the $TS_{max}$ by changing the external voltage to measure the change of RI. The corresponding detection range is 1.321-1.323. And the detection accuracy can be an order of magnitude lower than the original RI.

If the concentration of biomolecules continue to increase at this time, we will consider changing the thicknesses of PMMA and sodium layers. It can be seen from the above discussion that PMMA has influence on the resonance angle, so we can adjust it to obtain a larger RI detection range, such as Fig. 4(d)-(f). When RI changes greatly, the smaller PMMA thickness is selected. The sensitivity ($S_{d_{3}}={\triangle d_{3}}/{\triangle n_{4}}$) is $1.538$ $\mu$m/UIR ranged from 1.320-1.346, and the detection accuracy can select the same order of magnitude as the original RI. After evaluating the change of RI in different environments, three different measuring accuracies correspond to three different gears of the sensor. It is worth noting that the sensitivity of RI detection vicinity can also be adjusted by changing the thickness of sodium metal layer ($d_{2}$). Such as in the Fig. 4(f), when $d_{3}=100$ nm, the sensitivity $S_{\delta }$ is $7.8$ $\mu$m/RIU at $d_{2}=60$ nm, $34.1$ $\mu$m/RIU at $d_{2}=80$ nm and $152.7$ $\mu$m/RIU at $d_{2}=100$ nm. At present, the maturity of WVA technology development makes the measurement of TS very accurate. Here, we discussed that a wide range of RI change can be measured by changing three types of parameters and the sensitivity of the sensor can be adjusted by the thickness of sodium layer.

In order to more clearly indicate the advantages of this structure, the comparison of SHEL sensing performance of different SPR multilayered structures is shown in Table 1. It can be seen that our maximum sensitivity can reach 1844.9 $\mu$m/RIU without WVA. Sensitivity and sensing range are the key factors to evaluate a sensor. Most importantly, our structure has the advantages of larger RI measurement range and adjustable sensitivity. If WVA technology is used, the sensitivity of the structure will be greatly improved. In the future, we will consider other methods such as varying the number of graphene layers to improve our sensitivity.

Table 1. Comparison of SHEL sensing performance of different SPR multilayered structures

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Next, we consider the influence of Fermi energy and incident angle on the TS, as shown in Fig. 5(b). When the $\theta$ changes continuously, there will be a sharp mutation at $E_{\mathit {f} } = 0.517$ eV. Similarly, the distribution of TS with angle also conforms to the above discussion at the mutation point ($E_{\mathit {f} } = 0.517$ eV). This is because the resonance angle of this structure is concentrated in $71.1^{\circ }-71.4^{\circ }$, where the reflectivity $|r_{p}|^{2}$ is small, resulting in a large TS. Figure 5(a) represents the conductivity of graphene changing with the Fermi energy. At $E_{\mathit {f} } = 0.517$ eV, the imaginary part of the conductivity has the minimum value, and the real part is the critical point, resulting in a sudden change of the TS. In order to facilitate this phenomenon, Fig. 5(c) and Fig. 5(d) are drew to illustrate. It can be seen that $TS_{max}$ is only $71$ $nm$ when $E_{\mathit {f} } < 0.517$ eV, while the $TS_{max}$ reaches $315$ $nm$ if $E_{\mathit {f} } > 0.517$ eV. According to slab-based model of graphene [52], the complex effective electrical permittivity is $\varepsilon _{g}=1+i \sigma / \omega \varepsilon _{0} d_{g}$, and the RI is $n_{g } =\sqrt {\varepsilon _{g} }$. $d_{g}$ is the thickness of monolayer graphene, which is usually assumed to be 0.34 nm. The real part of conductivity corresponds to the imaginary part of the dielectric constant. When $E_{\mathit {f} } > 0.517$ eV, the real part of conductivity and the dielectric loss are 0, which makes lager TS.

 

Fig. 5. (a) represents the real part (red solid line) and imaginary part (black dotted line) of graphene conductivity varying with the Fermi energy respectively. (b), (c) and (d) depict the relationship of TS with Fermi energy and incident angle. Here $d_{2}, d_{3}=100$ $nm, n_{4}=1.32$.

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Moreover, changing the Fermi energy from 0 to 1 eV, the resonance angle is concentrated in a small range of $71.1^{\circ }-71.4^{\circ }$. Different metals and thicknesses have different resonance angles, and the measurement of resonance angle is of great significance to the study of SPR, especially in multilayered structures. For this reason, we can use the Fermi energy (gate voltage) to measure it, as shown in Fig. 5(d). As the Fermi energy increases, the resonance angle decreases. In the traditional Kretschmann configuration SPR sensing, the resonance angle can be determined by $\sqrt {\varepsilon _{p}} \sin \theta _{SPR}=\operatorname {Re}\left (\sqrt {\varepsilon _{m} \varepsilon _{s} /\left (\varepsilon _{m}+\varepsilon _{s}\right )}\right )$ [53], where $\varepsilon _{p}$, $\varepsilon _{m}$ and $\varepsilon _{s}$ represent the relative permittivity of prism, metal and sensing medium, respectively. However, it is complex to use the effective dielectric constant ($\varepsilon _{eff}$) instead of $\varepsilon _{m}$ to calculate the resonance angle in multilayered structures. Adjusting the Fermi energy to observe the zero point of TS is convenient and accurate. For example, when the Fermi energy is 0.8 eV, it is easy to find that the resonance angle is $71.128^{\circ }$. Thus, the function of resonance angle detection is obtained.

Based on the above discussion, there is a potential application of Fermi energy detection by adjusting the incident angle. We fixed $n_{4}= 1.32$, and the results are shown in Fig. 6. When the incident angle is increased from $71.11^{\circ }$, the measurement range move to the smaller Fermi energy, but the greater sensitivity. Because of the huge TS in $E_{\textit {f} } > 0.517$ eV, the range $0.517$ eV $<\mathit {E_{f}} < 1$ eV are only discussed here. Surprisingly, Fermi energy of the $TS_{min}$ is corresponding to the Fermi energy of the $TS_{max}$ after increasing the incident angle every 0.04 degrees. This not only brings great convenience to the measurement of TS, but also realizes the continuous detection of Fermi energy. Therefore, the TS can be detected by every $0.04^{\circ }$, the detecting with wide range of Fermi energy can be realized. Here, through the above discussion, the purpose of multi-functional detection of resonance angle and Fermi energy is achieved.

 

Fig. 6. The relationship between TS and Fermi energy at different incident angles. Other parameters are the same as Fig. 5.

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In this paper, we mainly theoretically studied the SHEL controllable RI sensing and functions of resonance angle and Fermi energy detection in sodium based multilayered structure. However, the nano-multilayered structure fabricated in the experiment will inevitably deviate from the theory, such as surface roughness and thickness deviation. Due to the active chemical properties of sodium and technical limit of thermo-assisted spin-coating process, the experimental process of nano film preparation and WVA technology are not discussed here. In the future work, the experimental research will be followed up as soon as possible.

4. Conclusion

In summary, the controllable RI sensing and multi-functional detection multilayered structure based on sodium are explored. It is found that the sodium has a great influence on the TS, while the thickness of PMMA layer has a large impact on the resonance angle. If these two thicknesses reach the appropriate parameters, it will reach the $TS_{max}$. Then, due to the great potential of symmetrical distribution of TS in controlling polarized light, traditional noble metals gold and silver are compared with sodium and found that sodium has obvious advantages in TS and sensitivity. Moreover, the RI variation of three different orders can be measured by changing the Fermi energy, PMMA and sodium parameters. Particularly, changing the thickness of sodium can adjust the sensitivity of the RI sensing. Compared with other SHEL sensing SPR multilayered structures, our structure has the advantages of wide range RI measurement and controllable sensitivity. In addition, the $TS_{max}$ has two values between $E_{\mathit {f} } < 0.517$ eV and $E_{\mathit {f} } > 0.517$ eV. At the same time, the resonance angle of this structure is concentrated around $71.1^{\circ }-71.4^{\circ }$ , and this phenomenon can be used as an accurate measurement of resonance angle. Finally, we can increase the incident angle from $71.11^{\circ }$ every $0.04^{\circ }$ to obtain a wide range of Fermi energy measurements. Therefore, our work also realizes the multi-functional detecting of resonance angle and Fermi energy. These studies may open the possibility for the development of high-sensitivity optical refractive index sensor and other physical quantities of precision measurement.