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Abstract
In this paper, a photonic spin Hall effect (PSHE) sensor for high-precision refractive index (RI) detection and graphene layer number detection is proposed. Numerical analysis is performed by the transfer matrix method. The graphene material is introduced into the layered topology to stimulate the generation of PSHE phenomenon, and both H polarization and V polarization displacements occur simultaneously. The effects of parameters such as chemical potential, relaxation time, and external temperature on the PSHE shift are also discussed. The displacement of H polarization can be used for RI detection, and the measurement range (MR), sensitivity (S), figure of merit (FOM), and detection limit (DL) are 1.1-1.5, 127.85 degrees/RIU, 2412, and 2.08×10−5, respectively. The superior sensing performance provides a theoretical possibility for the detection of solids, liquids, and gases. The shift characteristic of V polarization is appropriate for detecting the number of layers in graphene, with a MR and S of 1-9 layers and 4.54 degrees/layer. The impacts of dielectric loss on sensor performance are also considered. We hope that the proposed PSHE multifunctional sensor can improve a theoretical idea for novel sensor design.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nowadays, with the vigorous development of high-precision scientific research and industry, the requirements for highly advanced sensor technology are gradually rising. Whether it is in theoretical research or practical application, the sensor technology for single-function detection has been very mature, but the measurement conditions are more demanding and highly dependent on the stability of the external environment. In the face of a complex and changeable external environment, the measurement accuracy is greatly declined [1–3]. The multivariable detection sensors are qualified for identify multiple uncertain factors at the same time, and can more timely and effectively perceive changes in the external environment, so it has become a new research hotspot [4–7]. As an important parameter to characterize the physical properties of a medium, refractive index (RI) is of great significance for the development of material detection, biometrics, chemical industry and other fields to judge this property in a timely and accurate manner. Therefore, new methods for RI measurement have emerged in an endless stream in recent years [8]. Graphene is a honeycomb-shaped unparalleled two-dimensional structure composed of carbon atoms. Its carriers is manifested as a qualityless Dirac fermions, which produces many electrical properties such as anomalous quantum Hall effect, general conductivity and special Andrev reflection. In addition to electrical characteristics, it also has many charming optical characteristics [9]. Whether it is the theoretical or experimental level, it shows strong application potential in many fields such as optical absorption, transparent electrode, optical transition, and transformation of optical, and is also considered precious metal in the waveguide structure that may replace the sub-wavelength band. Many unique features are derived from the ultra-thin structure of graphene, so the separation and judgment of single-layer graphene are significant for the development of the graphene industry [10].
Thanks to the vulnerability of photonic spin Hall effect (PSHE) to the external environment, it has attracted attention in many sensor design mechanisms. PSHE refers to the phenomenon that when a linearly polarized beam is incident on the surface of a structure and reflected, the scattered light field spins apart due to the influence of different geometric phases of the left-handed and right-handed circularly polarized components of the beam. PSHE originates from the effective orbital coupling between the spin and the beam trajectory, which is highly dependent on the RI gradient of the medium. Initially, the spin splitting shift in the PSHE is very weak, usually only on the nanometer scale, making it difficult to observe directly. On account of the introduction of weak measurement techniques [11–13], the spin splitting displacement has been increased by several orders of magnitude and is relatively easy to detect. As such, it has been widely studied as an interesting optical phenomenon, such as optical physics [14], high energy physics [15], plasmonics [16], metasurface [17] and so on.
Remarkably, PSHE holds great promise for precision measurements due to its high dependence on RI gradients. Previous employments have explored its sensing properties [18–20], such as RI detection, graphene layer identification, and axial coupling detection of topological insulators. In 2018, Zhou et al [11]. proposed a RI sensor that uses weak measurement methods for detection. In this structure, based on the adsorption between graphene materials and biomolecules, a detection device with a measurement range (MR) of 1.33-1.335 and a sensitivity (S) of 1.088×105 µm/RIU was designed using surface plasmon resonance technology. Although the sensor has high S, its MR is too narrow and its resolution is rather low, which is not conducive to practical applications. In the same year, Sheng et al [21]. put forward a highly sensitive RI sensor by reducing the wave vector, and its S is as high as 3.16×10−5 RIU/µm. However, its MR is only three RI points, which is not a continuous range, meaning that this is not a healthy measurement method. In 2020, Zhang et al [22]. realized the measurement of low-concentration sodium chloride solution by using the magneto-optic PSHE effect. Based on the noreciprocal transmission characteristics of the magneto-optic effect, the sensor has a MR of 0-2% of the concentration of sodium chloride and a S of 2.9×104 μm/RIU. Although predecessors have proposed many sensor ideas based on the PSHE effect, they focus more on the detection of a single function.
In this paper, the movement in the angular position produced by PSHE, rather than the magnitude of the displacement, is used to detect the analyte, which to some extent reduces the risk of errors due to factors such as loss. The RI sensing and graphene layer number detection can be achieved by the H-polarization and and V-polarization. The dependence of different polarized PSHE on chemical potential, relaxation time and temperature is explored in detail. For the RI meaurement, the MR, S, figure of merit (FOM), and detection limit (DL) are 1.1-1.5, 127.85 degrees/RIU, 2412, and 2.08×10−5, respectively. For the graphene layer detection, the MR and S are 1-9 layers and 4.54 degrees/layer. The effects of dielectric loss on the sensor are also briefly considered, and a lower loss tangent is beneficial to improve FOM, reduce DL, and enhance the realization of high-precision sensors.
2. Structure design and simulation
The sensing structure exhibited in Fig. 1 consists of dielectric A, dielectric B (the analytes in the following text are set here), Graphene alternately. The RIs and thicknesses of dielectrics A and B are respectively nA = 2.45, dA = 1 µm and nB = 1.45, dB = 5.5 µm. The conductivity of the graphene material can be calculated by the Kubo equation [23,24]:
$\sigma _g^{{\mathop{\rm int}} er}$ and $\sigma _g^{{\mathop{\rm int}} ra}$ are the inter-band and intra-band conductivities of graphene, respectively:
Fig. 1. The proposed PSHE sensing structure and the linear polarization state of incident light and the transverse shifts of the left and right circularly polarized light component of reflected light. θi and θr represent the incident and reflection angles respectively. The light propagates in the + z direction.
Among them, ε0 is the vacuum dielectric constant, and dg = 0.34 nm signifies the thickness of the graphene layer.
The energy transition between layers is calculated using the transfer matrix method. The transmission matrix of each layer of medium can be described as [25]:
The energy transfer of the entire structure is expressed as:
The reflection coefficient is calculated as below:
When a beam of linearly polarized light is reflected on a structural surface with RI gradient, it splits into left-handed and right-handed circularly polarized light. The vertical and horizontal beam shifts of reflected light can be expressed as δV and δH. H polarization and V polarization represent the horizontal and vertical offset directions respectively [21,26]:
S, quality facter (QF), FOM, and DL are important parameters to evaluate a sensor, and an excellent sensor corresponds to powerful S, high QF, productive FOM, and diminutive DL. Corresponding definitions can be expressed as follows, where Δθ and Δna refer to the angle and RI change, while θT symbolizes PSHE resonant angle, and FWHM implies the full width half maximun [27–33].
3. Analysis and discussion
In our investigation, the frequency is set as 8.5 THz and the ratio of the amount of displacement to the incident wavelength λ for different polarizations is presented. As illustrated in Figs. 2(a) and (b), for the same μ value setting, both H polarization and V polarization exhibit prominent PSHE phenomena at the same time, but the resulting angular positions are deviated. When the value of μ extends from 0.1 eV to 0.4 eV, the PSHE of H polarization and V polarization are generated at 57.18 degrees and 64.61 degrees, 54.41 degrees and 59.44 degrees, 51.54 degrees and 54.74 degrees, 48.6 degrees and 50.37 degrees, respectively. The increase of μ pushes the position of the PSHE move towards the small angle, and the angle differences in the two polarization cases are 7.43 degrees, 5.03 degrees, 3.2 degrees and 1.77 degrees correspondingly, which means that the resonance angles in H polarization and V polarization are getting more and more near. Concurrently, the expansion of μ also forces the displacement to be gradually reduced. When the value of μ is 0.1 eV, the PSHE displacement under H polarization reaches a maximum of 580.22λ, and the value under V polarization is 6.99λ. However, if μ is changed to 0.4 eV, the corresponding displacements are 18.81λ and 1.32λ, respectively, and the degree of attenuation is very sharp. The peak is weakened and the width of the peak is widened, lessening the QF value, which is not conducive to the design of high-precision sensors.
Fig. 2. The effects of different μ on the PSHE phenomena of (a) H polarization and (b) V polarization.
Figure 3 highlights the adjustment of the PSHE phenomena under the two polarizations for different values of τ. In Fig. 3(a), for H polarization, when τ is 2 ps, 4 ps, 6 ps, and 8 ps, respectively, the size of the PSHE displacement is transfered to 146.21λ, 292.16λ, 438.16λ, and 582.74λ accordingly. And the angular positions are 57.174 degrees, 57.176 degrees, 57.176 degrees and 57.177 degrees, respectively. Remarkably, the enhancement of τ greatly improves the PSHE shift amount under H polarization, which is more favorable to be observed, and contributes to the improvement of QF. Interestingly, this change hardly affects the angular position. For Fig. 3(b), a similar effect also occurs in the case of V polarization. As τ aggrandizes from 2 ps to 8 ps at intervals of 2 ps, the displacement magnitudes are 1.69λ, 3.46λ, 5.23λ, 6.99λ, and the angular positions are 64.594 degrees, 64.604 degrees, 64.608 degrees, 64.609 degrees.
Fig. 3. The effects of different τ on the PSHE phenomena of (a) H polarization and (b) V polarization.
For graphene materials, temperature T is also an important factor for tuning its optical properties, and the influences of unstable temperature variation on device function are inevitable. In Fig. 4(a), the angular position of the PSHE under H polarization alters mildly when T transforms from 280 K to 320 K, shifting from 57.18 degrees to 57.17 degrees. For the V polarization in Fig. 4(b), the augment in T also contribute to a diminutive contractibility in the resonance angle, which is initially 64.63 degrees and then adjusted to 64.59 degrees. It can be observed that V polarization is slightly more affected by T than H polarization, but the overall remains relatively stable and is not significantly affected by T. It can also be seen from Fig. 4 that there is no visible change in the width of the resonance peak with the change of T, which means that the QF does not undergo a large adjustment during the whole process.
Fig. 4. The effects of different T on the PSHE phenomenon of (a) H-polarization and (b) V-polarization.
By exploring the effects of graphene related parameters on the PSHE phenomenon under different polarizations, it can be found that a fainter μ and a higher τ are beneficial to raise the peak value and decline the full width at half maximum, whereas the tuning effect of T on PSHE is awfully weak. Consequently, μ=0.1 eV, τ=8 ps, T = 300 K is chosen to premeditate the sensing performance, with the position of medium B as the analyte. As we know, the RIs of many liquids and solids, and even some gases, fall within the range of 1.1–1.5, so the study of high-performance sensors with ultra-high S and extremely low DL is in favour of perfecting related research. By comparison, H polarization with a abundant QF characteristic is selected to investigate RI detection. In Fig. 5(a), for the convenience of exploration, flve RI points are employed as typicality to lucubrate the sensing performance in this range. If the RI distribution takes 1.1, 1.2, 1.3, 1.4, and 1.5, the corresponding resonance angle positions are implemented to 13.46 degrees, 26.62 degrees, 38.05 degrees, 50.19 degrees, and 65.6 degrees, respectively. After linear fitting, S can reach 127.85 degrees/RIU, and the linearity R2 is 0.9964, these efficient indicators mean excellent sensing performance. In Fig. 5(b), through further numeration, it can be seen that the FOM values of the five points are 2412, 4566, 6728, 7990, and 6392, and the lower DL is as low as 2.08×10−5, 1.09×10−5, 7.43×10−6, 7.43×10−6, and 7.43×10−6. The analysis reveals that the minimum FOM value is 2412, and the maximum DL is 2.08×10−5, which is conducive to the detection of most conventional gases, liquids and solid substances. Similarly, QF is also an important index to evaluate the sensor performance. In Fig. 5(c), the QF values of the five analysis points are 253, 950, 2002, 3136, and 3280 respectively, with the lowest being 253 and the highest being 3280. If the structure is used to measure concrete objects, for example, when a liquid is put into the cavity to be measured and the resonance angle displayed is 41.316 degrees, it can be inferred from the sensing structure that its refractive index is 1.3198 and the liquid is pure water [34].
Fig. 5. (a) The RI falls within the linear fit range of 1.1-1.5, (b) the distribution circumstances of FOM and DL within the MR, (c) the distribution circumstances of QF within the MR.
Compared to the RI detection, the identification of graphene layers n does not require extremely low DL, hence, V polarization is taken into account. The proposed sensing structure can discriminate n in the range of 1-9 layers, and here we pick several points as representatives for analysis. In Fig. 6(a), when n takes the value of 1 layer, 3 layers, 5 layers, 7 layers, and 9 layers in turn, PSHE is generated at angular positions such as 64.61 degrees, 54.23 degrees, 45.39 degrees, 36.79 degrees, and 27.85 degrees. In the case of a linear fit, the S and linear fit accuracy of 4.54 degrees/layer and 0.9983 are depicted in Fig. 6(b). These clear metrics offer a new possibility to measure the number of layers in graphene.
Fig. 6. (a) Resonance angle response when the number of graphene layers changes, (b) the linear fitting equation.
The above studies are based on the case of no loss, and in the process of processing, the dielectric loss may be introduced, so the influence on the loss has to be discussed. Graphene itself is a lossy medium, and medium B is set as the analyte, so the dielectric loss of medium A needs to be concerned. In Figs. 7(a) and 7(b), the situation where the loss tangent is equal to 0.005 is explored. In the case of RIs of 1.1, 1.2, 1.3, 1.4, 1.5, respectively, the resonance angles shift to 13.42 degrees, 26.59 degrees, 38.04 degrees, 50.18 degrees, 65.61 degrees. Compared with the lossless condition, the resonance angle change is relatively weak. The fitting results show that the S value is 127.97 degrees/RIU and the R2 is 0.9963, which is also not significantly different from the lossless case. Further, the dilated loss forces the FOM and DL to attenuate. It can be seen from Fig. 7(b) that the FOM and DL of the selected five points are 633, 1218, 1706, 2064, 1640 and 7.9×10−5, 4.1×10−5, 2.9×10−5, 2.4×10−5, 3×10−5, severally. Although the index has dropped distinctly, DL still maintains the order of magnitude of 10−5, and still possesses utmostly high detection performance. In Figs. 7(c) and 7(d), the condition of tanα=0.001 is mined. During the process of increasing RI from 1.1 to 1.5, the resonance angles are 13.37 degrees, 26.56 degrees, 38.02 degrees, 50.17 degrees, and 65.62 degrees, respectively. S and R2 become 128.11 degrees/RIU and 0.9972 in turn. Compared with the case without loss, these two performance indicators are not affected much, or even improved a little. For the FOM and DL metrics, the values are tuned to 367, 703, 956, 1197, 963 and 1.3×10−4, 7.1×10−5, 5.2×10−5, 4.1×10−5, 5.1×10−5, respectively. It is very obvious that with the further extension of the loss, these two indicators have clearly fallen, but the FOM and DL can still be maintained above 367 and below 1.3×10−4. It can be seen that the accumulation of dielectric loss declines the accuracy and resolution of RI sensing to a certain extent, but does not alter the value of S and linearity much.
Fig. 7. For RI sensing, (a) the linear fit for tanα=0.0005, (b) the distribution of FOM and DL within the MR for tanα=0.0005, (c) the linear fit for tanα=0.001, (d) the distribution of FOM and DL within the MR for tanα=0.001.
In Fig. 8, the effects of loss tangent on graphene layers count measurements are explored. As exhibited in Figs. 8(a) and 8(c), when the value of tanα is 0.0005, with n of 1 layer, 3 layers, 5 layers, 7 layers, and 9 layers, singly, the resonance angles fall at the positions of 64.6 degrees, 54.24 degrees, 45.41 degrees, 36.73 degrees, and 27.81 degrees. The corresponding S and R2 values are 4.55 degrees/layer and 0.9984. In Figs. 8(b) and 8(d), under the premise that tanα is 0.001, the corresponding angular positions are 64.58 degrees, 54.26 degrees, 45.44 degrees, 36.72 degrees, and 27.76 degrees. At the same time, the two indicators S and R2 are adjusted to 4.55 degrees/layer and 0.9985. From the above phenomenon, it can be inferred that the loss within a certain range does not affect the basic performance of the sensor.
Fig. 8. For graphene layers sensing, (a) resonance angle response for tanα=0.0005, (b) resonance angle response tanα=0.001, (c) the linear fitting equation for tanα=0.0005, (d) the linear fitting equation for tanα=0.001.
4. Conclusion
In summary, in this paper, the tuning effects of chemical potential, relaxation time and temperature on graphene PSHE under different polarizations are first investigated. The results show that the increase of μ causes the resonance angle to move toward the small angle direction, and the displacement decreases gradually. The augment of τ is beneficial to improve the displacement and has little effect on the resonance angle. The impacts of T change on PSHE are relatively weak, and the angular position shift is not evident. The QF of PSHE under H polarization is more productive and thus applies to RI sensing. MR, S, minimum FOM, maximum DL are 1.1-1.5, 127.85 degrees/RIU, 2412, and 2.08×10−5, respectively. Appropriate dielectric loss will not have a significant influence on the S and MR of the sensor, but will cause a slight attenuation in FOM and DL. V polarization is suitable for judging the number of graphene layers, and the corresponding MR and S are 1-9 layers and 4.54 degrees/layer. The introduction of dielectric loss also poses little threat to detection performance. We hope that the proposed PSHE sensor can provide new ideas for investigating PSHE phenomena and expanding new sensor designs. In addition, to demonstrate the high performance characteristics of the proposed sensor, Table 1 is used as a comparison.
Table 1. The comparison of the proposed sensor and the works of the predecessors.
Funding
National Natural Science Foundation of China (61427801, 61771127, U1536123, U1536124).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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