Design and analysis of graphene- and germanium-based plasmonic probe with photonic spin Hall effect in THz frequency region for magnetic field and refractive index sensing

Abstract

In this work, we analyze the design of a graphene- and germanium-based plasmonic sensor with photonic spin Hall effect (PSHE) for detection of refractive index (RI) of a gas medium and magnetic field (B) applied to the graphene monolayer in THz frequency region. The PSHE phenomenon is studied in both conventional as well as modified weak measurements. The effect of gaseous medium thickness (d₄), transverse magnetic (TM) mode’s order, and amplified angle parameter (Δ) is studied on the sensor’s performance. Parameters such as sensitivity, resolution, and figure of merit have been considered for sensor’s performance evaluation. The results indicate that in the conventional weak measurements, for a TM₁ mode (with d₄ = 20 µm, B = 0, and Δ = 0.1°), an RI resolution of 2.32 × 10⁻¹² RIU is achievable for gas medium in the range 1–1.1 RIU. In the modified weak measurements, for a TM₃ mode (with d₄ = 100 µm, B = 0, and Δ = 0.1°), the RI resolution close to 1.39 × 10⁻¹⁰ RIU is achievable for gas sensing. The same sensor design was also studied for magnetic field sensing while keeping the value of gaseous medium RI (n₄) as 1. The results indicate that for a TM₁ mode (with d₄ = 20 µm and Δ = 0.1°), in the conventional weak measurements, a magnetic field resolution of 5.31 × 10⁻⁴ µT (i.e., 0.53 nT) is achievable for a range 0–1 T of B. Further, it is found that in contrast with the conventional case, the resolutions in the modified weak measurements are improved for large values of the Δ. Some of the results emerge better or comparable with the resolutions of RI and magnetic field measurement (5 × 10⁻⁹ RIU and 0.7 µT or 1.22 × 10⁻¹¹ RIU and 1.46 × 10⁻² µT) existing in the literature.

Introduction

The refractive index (RI) and magnetic field resolutions hold great importance (minimum detectable) in evaluating the sensing performance of a surface plasmon resonance (SPR) sensor. An RI resolution of 5 × 10⁻⁹ RIU was obtained by using a fiber-optic Fabry–Perot interferometer (FPI) sensor for monitoring of small changes in the composition of gases as described in Pevec and Donlagic (2018).

A sensor whose sensing element is a ferrofluid filled high-birefringence-photonic crystal fiber (PCF) inserted into a Sagnac loop was used to get magnetic field sensitivity of 1.073 nm/mT with a resolution of 1 µT for a magnetic field ranging from 10 to 40 mT (Zhao et al. 2016).

Recently, the transverse spin-dependent shift (SDS) of the horizontal photonic spin Hall effect (PSHE) at a given frequency (5 THz) has been considered to analyze a graphene-based plasmonic sensor with four layers (germanium, dielectric, graphene, and gaseous medium) for detection of RI of a gas medium and magnetic field applied to the graphene layer (Popescu et al. 2022). PSHE refers to reflection and splitting of linearly polarized incident light into left circularly polarized (LCP) and right circularly polarized (RCP) light due to RI gradient prevailing in the given optical structure. The strong plasmonic properties of graphene monolayer are of great significance which can be exploited for the sensing purpose in THz frequency region. In the conventional weak measurements, an RI resolution of 1.22 × 10⁻¹¹ RIU was achievable for gas medium (when gas RI is changed from 1 to 1.1 RIU) (for a magnetic field B = 0 T and an amplified angle Δ = 0.1°). Further, in the conventional weak measurements, a magnetic field resolution of 1.46 × 10⁻² µT (gaseous medium RI taken as 1) was reported corresponding to magnetic field range of 0–0.1 T (for an amplified angle Δ = 0.1°). Also, in the modified weak measurements, a RI (magnetic field) resolution of 6.24 × 10⁻¹⁷ RIU (3.59 × 10⁻⁶ µT) was reported to be achievable (for an amplified angle Δ = 5.730°). These types of sensors could be extremely useful in early detection of airborne viruses such as SARS-Cov-2 and also in the early detection of radiation leakage in nuclear reactors (Kumar et al. 2022).

In another recent paper (Popescu et al. 2021), a plasmonic sensor with three (silicon, gold and magnetic fluid) layer waveguide structure was analyzed by using the transverse SDS of the horizontal PSHE in the waveguide at a given wavelength (1557 nm) for a TM mode. A magnetic field resolution of the order of 1.50 × 10⁻⁶ Oe was reported in the conventional weak measurements for an amplified angle Δ = 0.1° and 7.80 × 10⁻⁸ Oe in the modified weak measurements for Δ = 28.65°.

However, it is expected that by using additional layers of the concerned materials and/or manipulating their locations in the sensor design might bring reasonable improvement in the sensor’s performance. In this paper, we calculate the amplitude reflection coefficients by using the transfer matrix and guided-wave surface plasmon resonance methods. Then we analyze the influence of a supplementary germanium layer of a graphene-based plasmonic sensor with five layers (germanium, dielectric, graphene, gaseous medium, and germanium) for detection of the RI of a gas medium and magnetic field applied to the graphene monolayer at 5 THz frequency. The performance is closely analyzed in terms of the concerned performance parameters.

Plasmonic sensor with photonic spin Hall effect

Figure 1 shows the schematic diagram of the plasmonic sensor with PSHE with five layers-Ge prism, organic layer, graphene, gas medium and outside medium consisting of Ge.

Fig. 1.

Schematic of an optical structure with five layers (germanium, organic layer, graphene, gas medium and germanium) as a plasmonic sensor based on PSHE, where δ₋^H and δ₊^H are the transverse displacements of LCP and RCP components of light, respectively. The gaseous medium flows in the gas sensing layer from the left part

The RI values of the layers, for a fixed frequency of 5 THz, are listed in Table 1:

Table 1.

Optical parameters of the concerned layers in the sensor design (Popescu et al. 2022)

Layer	Material	RI	Thickness
1	Germanium	n1 = 4	–
2	Organic layer	n2 = 1.5	51 nm
3	Graphene	n3 = 5.189555 + 108.627427i at B = 0 T	0.34 nm
		n3 = 5.307784 + 109.1106197i at B = 1 T
4	Gas medium	n4 (varied from 1 RIU to 1.1 RIU)	d4 (varied: 1–200 µm)
5	Germanium	n5 = 4	–

The RI gradient plays the role of electric potential gradient. The real part of graphene RI is greater than the RI of Ge, and it ensures that the light is confined near the graphene layer in this plasmonic structure (Popescu et al. 2022). It should be specifically noted that the RI of graphene monolayer (n₃) is dependent on B incident perpendicular to the graphene surface (Popescu et al. 2022) as per following expression:

$$\begin{array}{cc}\hfill n_3& =\sqrt{{\epsilon }_g}=\sqrt{1+\frac{i{\sigma }_g}{\omega t_g{\epsilon }_0}}=\sqrt{1+\frac{i{\sigma }_0}{\omega t_g{\epsilon }_0}\times \frac{i(\omega +\frac{i}{\tau })}{{\left(\omega ,+,\frac{i}{\tau }\right)}^2-{\omega }_c^2}}\hfill \\ \hfill & =\sqrt{1+\frac{i{\sigma }_0\tau }{\omega t_g{\epsilon }_0}\times \frac{(1-i\omega \tau )}{{({\omega }_c\tau )}^2+{(1-i\omega \tau )}^2}},\hfill \\ \hfill \end{array}$$ 1

In Eq. (1), ε_g is the relative permittivity (dielectric constant) of graphene monolayer, σ_g is the conductivity of graphene, $\omega =2\pi \nu$ is the angular frequency, ν is the frequency, t_g = 0.34 nm is the effective thickness of graphene monolayer, ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of vacuum, ${\sigma }_0=e^2{E}_F/(\pi {ħ}^2)$, σ₀₀ = σ₀τ is a static conductivity without magnetic field, E_F = 0.3 eV is the Fermi energy, $\tau =\mu E_F/(e{\text{v}}_{\text{F}}^2)$ is the carrier relaxation time, μ = 1 m²/Vs is the carrier mobility, v_F = 9.5 × 10⁻⁵ m/s is the Fermi velocity and ${\omega }_c=eB{\text{v}}_{\text{F}}^2/E_F$ is the cyclotron frequency of electrons in graphene. At a frequency of ν = 5 THz the energy of the incident radiation of hν = 0.021 eV is smaller than a Fermi energy E_F = 0.3 eV. Also, the Fermi energy (0.3 eV) is larger than the thermic energy (k_BT = 0.025 eV) at the room temperature T = 293.15 K where k_B is the Boltzmann constant. In this case the interband conductivity (1.28 × 10⁻⁷–1.34 × 10⁻⁶i) is negligible, and the intraband conductivity (1.07 × 10⁻⁴ + 1.11 × 10⁻³i) is simplified.

The transverse SDS of two TM spin Hall effect components are defined as follows as per Popescu et al. (2022), Zhou et al. (2018):

$${\delta }_{\pm }^H=\mp \frac{\lambda }{2\pi }\left[1+\frac{\left|r_s\right|}{\left|r_p\right|}\right]cot\theta$$ 2

where r_p and r_s are the Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively, and θ is the incident angle. In this work, only the component δ_-^H of the horizontal polarization is considered. The corresponding amplified SDS in the conventional weak measurements is (Popescu et al. 2022; Zhou et al. 2018):

$$A{\delta }_{-}^H={\delta }_{-}^H\frac{z_r}{z_L}cot\mathrm{\Delta },$$ 3

where z_r (250 mm) is the z coordinate in the reflected light, $z_L=\frac{1}{2}k{w}^2$ is the Rayleigh length, k is the wave-vector, w is the beam waist (10 µm in the focused incident light), and Δ is the post-selected (amplified) angle (Popescu et al. 2022; Zhou et al. 2018). In the modified weak measurements (Popescu et al. 2022; Zhou et al. 2012), the amplified shift is:

$$A^m{\delta }_{-}^H=\frac{z_r(2k\left|r_p\right|z_L(\left|r_p\right|+\left|r_s\right|)+\left|{(\partial r_p/\partial \theta )}^2\right|)sin(2\mathrm{\Delta })cot\theta }{2k{z}_L{(\left|r_p\right|+\left|r_s\right|)}^2{cos}^2\mathrm{\Delta }{cot}^2\theta +4{(k\left|r_p\right|z_{L}sin\mathrm{\Delta })}^2}$$ 4

If $2k\left|r_p\right|z_L(\left|r_p\right|+\left|r_s\right|)>>\left|{(\partial r_p/\partial \theta )}^2\right|$ and $2k{z}_L{(\left|r_p\right|+\left|r_s\right|)}^2{cos}^2\mathrm{\Delta }{cot}^2\theta <<4{(k\left|r_p\right|z_{L}sin\mathrm{\Delta })}^2$, the amplified modified shift $A^m{\delta }_{-}^H$ can be replaced with amplified conventional shift $A{\delta }_{-}^H$.

Amplitude reflection coefficients (transfer matrix method)

In the transfer matrix method (Sharma et al. 2019; Popescu 2019a), the amplitude reflection coefficients (r_s for TE modes and r_p for TM modes) for a device with N = 5 layers are given by

$$\begin{array}{cc}\hfill r_s& =\frac{(M_{11}^s+M_{12}^s{q}_5^s)q_1^s-(M_{21}^s+M_{22}^s{q}_5^s)}{(M_{11}^s+M_{12}^s{q}_5^s)q_1^s+(M_{21}^s+M_{22}^s{q}_5^s)}\hfill \\ \hfill r_p& =\frac{(M_{11}^p+M_{12}^p{q}_5^p)q_1^p-(M_{21}^p+M_{22}^p{q}_5^p)}{(M_{11}^p+M_{12}^p{q}_5^p)q_1^p+(M_{21}^p+M_{22}^p{q}_5^p)},\hfill \\ \hfill \end{array}$$ 5

where

$$M^s=M_2^s{M}_3^s{M}_4^s=\left(\begin{array}{cc}{M}_{11}^s& M_{12}^s\\ M_{21}^s& M_{22}^s\end{array}\right),M^p=M_2^p{M}_3^p{M}_4^p=\left(\begin{array}{cc}{M}_{11}^p& M_{12}^p\\ M_{21}^p& M_{22}^p\end{array}\right),$$ 6

$$\begin{array}{cc}\hfill M_j^s& =\left(\begin{array}{cc}cos({\beta }_j)& -\frac{isin({\beta }_j)}{q_j^s}\\ -i{q}_j^{s}sin({\beta }_j)& cos({\beta }_j)\end{array}\right),\hfill \\ \hfill M_j^p& =\left(\begin{array}{cc}cos({\beta }_j)& -\frac{isin({\beta }_j)}{q_j^p}\\ -i{q}_j^{p}sin({\beta }_j)& cos({\beta }_j)\end{array}\right),\hfill \\ \hfill \end{array}$$ 7

where j = 2,3,4

$$q_1^s=\sqrt{n_1^2-n_1^2{sin}^2({\theta }_1)},q_1^p=\frac{q_1^s}{n_1^2},$$ 8

$$q_j^s=\sqrt{n_j^2-n_1^2{sin}^2({\theta }_1)},q_j^p=\frac{q_j^s}{n_j^2},{\beta }_j=\frac{2\pi d_j}{\lambda }\sqrt{n_j^2-n_1^2{sin}^2({\theta }_1)}$$ 9

where j = 2,3,4.

$$q_5^s=\sqrt{n_5^2-n_1^2{sin}^2({\theta }_1)},q_5^p=\frac{q_5^s}{n_5^2}$$ 10

The corresponding intensity reflection coefficients are:

$$R_s={\left|r_s\right|}^2,R_p={\left|r_p\right|}^2$$ 11

The corresponding output powers are:

$$P_s=R_s^{\frac{L}{Dtan({\theta }_1)}},P_p=R_p^{\frac{L}{Dtan({\theta }_1)}},$$ 12

where L is the sensing length, D is the width of the core layers with incident angle θ₁ in optical structure (L/D = 25). The corresponding power losses (in dB) are:

$$P{L}_s=10Log\frac{1}{P_s},P{L}_p=10Log\frac{1}{P_p}$$ 13

Amplitude reflection coefficients (guided-wave surface plasmon resonance configuration)

Another similar variant to calculate the amplitude reflection (Fresnel) coefficients (r_p and r_s) is to use the guided-wave surface plasmon resonance (GWSPR) configuration (Xiang et al. 2017; Wang et al. 2019):

$$r_s=\frac{r_{12}^s+r_{2345}^{s}exp(i{\varphi }_2)}{1+r_{12}^s{r}_{2345}^{s}exp(i{\varphi }_2)},r_p=\frac{r_{12}^p+r_{2345}^{p}exp(i{\varphi }_2)}{1+r_{12}^p{r}_{2345}^{p}exp(i{\varphi }_2)},$$ 14

where

$$\begin{array}{cc}\hfill r_{j(j+1)}^s& =\frac{n_{j}cos{\theta }_j-\sqrt{n_{(j+1)}^2-n_j^2{sin}^2{\theta }_j}}{n_{j}cos{\theta }_j+\sqrt{n_{(j+1)}^2-n_j^2{sin}^2{\theta }_j}},\hfill \\ \hfill r_{j(j+1)}^p& =\frac{n_{(j+1)}^{2}cos{\theta }_j-n_j\sqrt{n_{(j+1)}^2-n_j^2{sin}^2{\theta }_j}}{n_{(j+1)}^{2}cos{\theta }_j+n_j\sqrt{n_{(j+1)}^2-n_j^2{sin}^2{\theta }_j}}\hfill \\ \hfill \end{array}$$ 15

where j = 1,2,3,4

$$r_{345}^s=\frac{r_{34}^s+r_{45}^{s}exp(i{\varphi }_4)}{1+r_{34}^s{r}_{45}^{s}exp(i{\varphi }_4)},r_{345}^p=\frac{r_{34}^p+r_{45}^{p}exp(i{\varphi }_4)}{1+r_{34}^p{r}_{45}^{p}exp(i{\varphi }_4)},$$ 16

$$r_{2345}^s=\frac{r_{23}^s+r_{345}^{s}exp(i{\varphi }_3)}{1+r_{23}^s{r}_{345}^{s}exp(i{\varphi }_3)},r_{2345}^p=\frac{r_{23}^p+r_{345}^{p}exp(i{\varphi }_3)}{1+r_{23}^p{r}_{345}^{p}exp(i{\varphi }_3)},$$ 17

The corresponding phase term can be given as:

$${\varphi }_j=\frac{4\pi }{\lambda }{n}_j{d}_{j}cos{\theta }_j$$ 18

$${\theta }_j=arcsin\frac{n_{(j-1)}sin{\theta }_{(j-1)}}{n_j}$$ 19

where j = 2,3,4.

Sensor’s performance parameters

In the angular interrogation method (Popescu 2019b; Popescu and Sharma 2019, 2020; Sharma et al. 2020), the wavelength is kept constant and the angle of incidence is varied and a sharp dip (maximum of the power loss) appears at a resonance incidence angle θ₁ in the reflectivity. The resonance incidence angle is dependent on the RI of the sensing medium. The sensing performance of the sensor is primarily determined by the angular sensitivity (S_θ) calculated for an increase of n_a with δn_a:

$$S_{\theta }=\frac{\delta {\theta }_{\mathit{res}}}{\delta n_a},$$ 20

the maximum of the power sensitivity (S_PL) calculated for an increase of n_a with δn_a,

$$S_{\mathit{PL}}=\frac{\delta PL}{\delta n_a},$$ 21

the figure of merit (FOM):

$$FOM=\frac{\delta {\theta }_{\mathit{res}}}{\delta n_a\delta {\theta }_{0.5}},$$ 22

where δθ_res is the shift in the resonance angle corresponding to the variation δn_a in the analyte RI, δθ_0.5 is the angular width of resonance spectrum, PL_a is the power loss at n_a + δn_a, and PL is power loss corresponding to reference analyte n_a.

The sensitivity (S_n in µm/RIU) of gaseous RI measurement is defined as the ratio of the difference in the optimal amplified shift to the change in sensing (gas) medium RI (Srivastava et al. 2021; Zhou et al. 2018; Prajapati 2021):

$$S_n=\frac{\mathrm{\Delta }(A{\delta }_{-}^H)}{\delta n_4}=\frac{\left|A^{{}^{\prime }}{\delta }_{-}^H-A{\delta }_{-}^H\right|}{\delta n_4}$$ 23

where Aδ₋^H and A′δ₋^H are amplified SDS for a given value of the magnetic field and analyte RI values of n₄ = 1 and or n_4a = 1.1, (i.e., δn₄ = 0.1 RIU) respectively. The corresponding RI resolution (in RIU) is defined as:

$$R_n=\frac{\delta d}{S_n},$$ 24

where it is assumed that the resolution of spatial measurement δd is 0.2 µm. This value of δd corresponds to the resolution of a commercial laser sensor (Laser thickness gauge 2021).

The magnetic field sensitivity (S_B in µm/T) is defined for a given value of n₄ = 1 as the ratio of difference in the optimal amplified shift to the change in the magnetic field (which basically leads to a variation in graphene layer RI) as:

$$S_B=\frac{\mathrm{\Delta }(A{\delta }_{-}^H)}{\delta B}=\frac{\left|A^{{}^{\prime }}{\delta }_{-}^H-A{\delta }_{-}^H\right|}{\delta B}$$ 25

The magnetic field values are varied between B = 0 T and B = 1 T. The corresponding magnetic field resolution (in T) is defined as

$$R_B=\frac{\delta d}{S_B},$$ 26

Here, it is assumed (and as mentioned earlier) that the resolution of spatial measurement (δd) is 0.2 µm.

Results and discussion

Occurrence of SPR in the proposed sensor design

Keeping in view that the RI sensitivity is affected by the thickness of the sensing layer (d₄), we have tried to investigate how the variation of d₄ affects the sensor’s performance. In this context, Fig. 2 shows the variation of |r_p| and PL with incident angle θ in optical structure corresponding to a TM₁ single mode for d₄ = 20 µm, n₄ = 1 (a), n_4a = 1.1 (b), and B = 0 T.

Fig. 2.

Variation of absolute value of a Fresnel reflection coefficient |r_p|, and b power loss PL with incident angle θ in optical structure corresponding to a TM₁ mode for d₄ = 20 µm, n₄ = 1, n_4a = 1.1 and B = 0 T

The above figure shows that the resonance takes place at θ = 14.1288° and PL = 3142.906 dB for n₄ = 1. Further, for n_4a = 1.1, the resonance occurs at θ = 15.4724° and PL = 2733.183 dB. Therefore, for B = 0 T, the angular sensitivity, S_θ, under the angular interrogation method turns out to be 13.4353° RIU⁻¹ and the corresponding resolution, i.e., R_θ is 7.44 × 10⁻⁶ RIU while assuming that the standard resolution (δθ_a) available for angular measurement is 0.0001° (Sharma and Nagao 2014). The corresponding power loss sensitivity, S_PL, is 4097.233 dB/RIU and the corresponding resolution, R_PL, is 2.44 × 10⁻⁶ RIU while assuming that the standard power loss resolution (δPL_a) available is 0.01 dB (Gao and Jiang 2013). Further, the angular width of the resonance spectrum is δθ_0.5 = 0.6107°, and the corresponding FOM is 21.999 RIU⁻¹.

Influence of variation in TM mode and d₄ on sensing performance

Getting ahead with the analysis, Fig. 3 shows the variation of PL with incident angle θ in optical structure corresponding to TM₁, TM₂ and TM₃ modes for d₄ = 100 µm, n₄ = 1 and B = 0 T.

Fig. 3.

Variation of power loss PL with incident angle θ in optical structure corresponding to, TM₁, TM₂ and TM₃ modes for d₄ = 100 µm, n₄ = 1 and B = 0 T

Above figure shows that for n₄ = 1 and B = 0 T, the resonance (i.e., power loss peak) for TM₁, TM₂ and TM₃ modes takes place at θ = 13.9128° (PL = 3205.902 dB), 11.6012° (PL = 6126.030 dB), and 6.3768° (PL = 11,410.844 dB), respectively. Further, for n_4a = 1.1, the resonance corresponding to TM₃ mode occurs at θ = 9.1974° and PL = 7863.956 dB. Therefore, for B = 0 T, the angular sensitivity (S_θ) under angular interrogation method turns out to be 28.2057° RIU⁻¹ and the corresponding resolution (R_θ) is 3.54 × 10⁻⁶ RIU. The above values suggest that the power loss sensitivity (S_PL) and resolution (R_PL) corresponding to TM₃ mode are 35,468.881 dB/RIU and 2.82 × 10⁻⁷ RIU, respectively. Further, δθ_0.5 = 0.1793° for PL curve corresponding to TM₃ mode, and the corresponding FOM comes out to be 157.279 RIU⁻¹.

So, the comparison between results presented in Figs. 2 and 3 reveal that if we increase d₄ from 20 µm for a TM₁ mode to 100 µm for a TM₃ mode, the angular width of resonance spectrum decreases considerably from 0.6107° to 0.1793° leading to nearly 7.5 times increase in FOM from 21.999 to 157.279 RIU⁻¹. The above is a significant result in terms of sensing performance enhancement. Taking cue from this result, if one considers even higher order mode (e.g., TM₆) with greater d₄, the FOM is expected to get further improved.

In order to have a broader outlook of the above discussion, Table 2 summarizes the values of PL (dB), S_θ (°/RIU), R_θ (RIU), S_P (dB/RIU), R_P (RIU), δθ_0.5 (°) and FOM (RIU⁻¹) for a TM₁ mode when d₄ is 1.2 µm, 10 µm, 20 µm, 30 µm. The same table also contains the corresponding values for a TM₃ mode when d₄ = 100 µm, and for a TM₆ mode when d₄ = 200 µm.

Table 2.

Values of relevant sensor parameters for a TM₁ mode for 4 different values of d₄ (1.2–30 µm). For comparison purpose, all these sensor parameters are also listed for a TM₃ mode when d₄ = 100 µm and for a TM₆ mode when d₄ = 200 µm for B = 0 T

Mode	d4	PL	Sθ	Rθ	SP	RP	δθ0.5	FOM
TM1	1.2	5861.7	14.63	6.83 × 10−6	1.02 × 104	9.84 × 10−7	1.27	11.49
TM1	10	3734.6	13.55	7.38 × 10−6	4.86 × 103	2.06 × 10−6	0.81	16.64
TM1	20	3142.9	13.44	7.44 × 10−6	4.10 × 103	2.44 × 10−6	0.61	22.00
TM1	30	2829.6	13.37	7.48 × 10−6	3.57 × 103	2.80 × 10−6	0.53	25.33
TM3	100	11410.8	28.21	3.55 × 10−6	3.55 × 104	2.82 × 10−7	0.18	157.28
TM6	200	11502.5	28.28	3.54 × 10−6	3.60 × 104	2.78 × 10−7	0.09	314.38

The values contained in Table 2 clearly indicate that for TM₁ mode, the FOM improves as the value of d₄ increases. This is due to a significant decrease in δθ_0.5, which leads the overall FOM to increase even though the sensitivity decreases with an increase in d₄. Further, for even greater magnitude of d₄ (100 nm, 200 nm) and for higher order modes (TM₃, TM₆), there is a highly significant increase in the sensor’s FOM. More precisely, the FOM reaches as high as 314.38 RIU⁻¹ for TM₆ mode with d₄ = 200 nm.

Analysis of sensing performance with PSHE (Conventional weak measurements)

Figure 4 shows the simulated variation of initial SDS (δ₋^H) with θ in the optical structure corresponding to a TM₁ mode for d₄ = 20 µm and when B = 0 T. Figure 4 contains the SDS vs. θ curves for two values of n₄ (1 and 1.1 RIU). For n₄ = 1, the maximum of SDS (δ₋^H) = 1416.302 µm at θ = 14.1293°. The maximum of δ₋^H becomes 1101.629 µm at θ = 15.4731° for n₄ = 1.1 RIU. Thus, the RI sensitivity at B = 0 T is [δ₋^H (n₄ = 1) − δ_-^H (n₄ = 1.1)]/δn_a = 3146.735 µm/RIU.

Fig. 4.

Variation of initial SDS (δ₋^H) with incident angle θ in optical structure corresponding to a TM₁ single mode for d₄ = 20 µm, n₄ = 1, n₄ = 1.1 and B = 0 T

Extending the analysis for a structure with d₄ = 100 µm, n₄ = 1, B = 0 T and a TM₃ mode, it is observed that the maximum of δ_-^H is 544.728 µm at θ = 6.3780° for n₄ = 1 RIU, which changes to 699.602 µm at θ = 9.1980° for n₄ = 1.1 RIU. These values indicate that the RI sensitivity for this very structure is 1548.74 µm/RIU, which is smaller than the sensitivity (3146.735 µm/RIU) discussed above.

As discussed in the previous sections, it is also possible to resort to an amplified SDS in order to seek an improvement in the sensing performance of the concerned PSHE structure. In this context, Fig. 5 shows the variation of amplified SDS, i.e., Aδ₋^H (for n₄ = 1 and for n_4a = 1.1) with θ for a TM mode in conventional weak measurements (Δ = 0.1°) corresponding to B = 0 T.

Fig. 5.

Variation of amplified SDS (Aδ₋^H) with incident angle θ in optical structure corresponding to a TM₁ single mode in conventional weak measurements (Δ = 0.1°) for d₄ = 20 µm, n₄ = 1, n₄ = 1.1 and B = 0 T

Above plots indicate that the peak value of Aδ₋^H at n₄ = 1 is 3.87 × 10¹⁰ µm and it corresponds to θ = 14.1293°. Similarly, the peak value of Aδ₋^H at n_4a = 1.1 is 3.01 × 10¹⁰ µm (at θ = 15.473°). Thus, sensitivity is 8.60 × 10¹⁰ µm/RIU and corresponding resolution is 2.32 × 10⁻¹² RIU.

Analyzing the magnetic field sensing, it is found that Aδ₋^H = 3.83 × 10¹⁰ µm (at θ = 14.1301°) for B = 1 T and n₄ = 1. Thus, the magnetic field sensitivity turns out to be 3.77 × 10² µm/µT ($\mathrm{\Delta }B$ = 1 T) while the corresponding resolution is 5.31 × 10⁻⁴ µT.

For another structure with d₄ = 100 µm, n₄ = 1, B = 0 T, Δ = 0.1° and a TM₃ mode, in conventional weak measurements, the peak value of Aδ₋^H is 1.49 × 10¹⁰ µm at θ = 6.3780°. The peak value of Aδ₋^H becomes 1.91 × 10¹⁰ µm (at θ = 9.1980°) for n₄ = 1.1. Thus, sensitivity is 4.23 × 10¹⁰ µm/RIU and corresponding resolution is 4.72 × 10⁻¹² RIU. For B = 1 T and n₄ = 1, Aδ₋^H = 1.50 × 10¹⁰ µm. Thus, the magnetic field sensitivity turns out to be 1.45 × 10² µm/µT ($\mathrm{\Delta }B$ = 1 T) while the corresponding resolution is 1.38 × 10⁻³ µT.

Analysis of sensing performance with PSHE (Modified weak measurements)

A similar analysis is carried out for the amplified SDS, i.e., Aδ₋^H (for n₄ = 1 and n_4a = 1.1) with θ for a TM₁ mode (Fig. 6) under modified weak measurements (Δ = 0.1°) corresponding to B = 0 T. The corresponding maximum values of Aδ₋^H are 2.11 × 10⁵ µm (at θ = 14.1594° for n₄ = 1) and 1.82 × 10⁵ µm (at θ = 15.5225° for n_4a = 1.1). Thus, sensitivity is 2.86 × 10⁵ µm/RIU and corresponding resolution is 6.99 × 10⁻⁷ RIU. For B = 1 T and n₄ = 1, Aδ₋^H = 2.10 × 10⁵ µm (at θ = 14.1605°). Thus, the magnetic field sensitivity is 1.07 × 10⁻⁴ µm/µT (δB = 1 T) while the corresponding resolution is 1.87 × 10³ µT = 1.87 mT.

Fig. 6.

Variation of amplified SDS (Aδ₋^H) with incident angle θ in optical structure corresponding to a TM₁ single mode in modified weak measurements (Δ = 0.1°) for d₄ = 20 µm, n₄ = 1, n₄ = 1.1 and B = 0 T

For another structure with d₄ = 100 µm, n₄ = 1, B = 0 T, Δ = 0.1° and a TM₃ mode, in modified weak measurements, the maximum value of Aδ₋^H is 5.49 × 10⁸ µm (at θ = 6.3621° for n₄ = 1) and 4.06 × 10⁸ µm (at θ = 9.1713° for n_4a = 1.1). Thus, sensitivity is 1.44 × 10⁹ µm/RIU and corresponding resolution is 1.39 × 10⁻¹⁰ RIU.

For B = 1 T and n₄ = 1, Aδ₋^H = 5.43 × 10⁸ µm, which leads to a magnetic field sensitivity is 6.3941 µm/µT ($\mathrm{\Delta }B$ = 1 T) while the corresponding resolution is 3.13 × 10⁻² µT = 3.13 × 10⁻⁵ mT.

Table 3 summarizes the values of S_n^c (µm/RIU), R_n^c (RIU) with the conventional weak measurements and S_n^m (µm/RIU), R_n^m (RIU) with the modified weak measurements for a TM₁ mode when d₄ is varied between1.2 µm and 30 µm. The same table also contains the corresponding values for a TM₃ mode when d₄ = 100 µm, and for a TM₆ mode when d₄ = 200 µm for Δ = 0.1° and B = 0 T.

Table 3.

Values of S_n^c (µm/RIU) and R_n^c (RIU) with the conventional weak measurements, and S_n^m (µm/RIU) and R_n^m (RIU) with the modified weak measurements for a TM₁ mode when d₄ is 1.2 µm, 10 µm, 20 µm, 30 µm, for a TM₃ mode when d₄ = 100 µm and for a TM₆ mode when d₄ = 200 µm for Δ = 0.1° and B = 0 T

Mode	d4	Snc	Rnc	Snm	Rnm
TM1	1.2	2.76 × 1012	7.26 × 10−14	1.16 × 104	1.73 × 10−5
TM1	10	1.78 × 1011	1.12 × 10−12	7.33 × 104	2.73 × 10−6
TM1	20	8.60 × 1010	2.32 × 10−12	2.86 × 105	6.99 × 10−7
TM1	30	5.42 × 1010	3.69 × 10−12	6.99 × 105	2.86 × 10−7
TM3	100	4.23 × 1010	4.72 × 10−12	1.44 × 109	1.39 × 10−10
TM6	200	4.19 × 1010	4.77 × 10−12	5.80 × 109	3.45 × 10−11

Table 3 shows that for larger value of d₄, the S_n^c decreases while S_n^m increases. This, in effect, leads to achieve finer resolution of RI measurement for smaller d₄ with PSHE under conventional weak measurements. On the other hand, the finer RI resolution is achieved for larger d₄ with PSHE under modified weak measurements.

In this sequence, Table 4 enlists the values of sensitivity S_B^c (µm/µT), and resolution R_B^c (µT) with the conventional weak measurements and S_B^m (µm/µT), R_B^m (µT) with the modified weak measurements of magnetic field for a TM₁ mode when d₄ is varied between 1.2 µm and 20 µm. The same table also contains the corresponding values for a TM₃ mode when d₄ = 100 µm, and for a TM₆ mode when d₄ = 200 µm for Δ = 0.1° and n₄ = 1.

Table 4.

Values of S_B^c (µm/µT), R_B^c (µT) with the conventional weak measurements and S_B^m (µm/µT), R_B^m (µT) with the modified weak measurements for a TM₁ mode when d₄ is 1.2 µm, 10 µm, 20 µm for a TM₃ mode, when d₄ = 100 µm and for a TM₆ mode when d₄ = 200 µm for Δ = 0.1° and n₄ = 1

Mode	d4	SBc	RBc	SBm	RBm
TM1	1.2	4.04 × 104	4.96 × 10−6	8.05 × 106	2.48 × 10−4
TM1	10	8.31 × 102	2.41 × 10−4	3.67 × 105	5.45 × 10−3
TM1	20	3.77 × 102	5.31 × 10−4	1.07 × 104	1.87 × 10−3
TM3	100	1.46 × 102	1.38 × 10−3	6.39	3.13 × 10−2
TM6	200	1.42 × 102	1.41 × 10−3	25.4	7.86 × 10−3

Table 4 indicates that for larger value of d₄, the S_B^c decreases while S_B^m increases. This, in effect, leads to achieve finer resolution of magnetic field measurement for smaller d₄ with PSHE under conventional weak measurements. On the other hand, the finer magnetic field resolution is achieved for larger d₄ with PSHE under modified weak measurements.

From Tables 3 and 4, it is also apparent that a higher order mode (e.g., TM₆) is favorable for achieving greater sensitivity and finer resolution of magnetic field under the modified weak measurements.

Influence of variation in Δ along with TM mode order and d₄ on sensing performance

We have also simulated other different variants (in terms of extended variations in Δ along with above-discussed sets of TM modes and d₄) of the proposed sensor design in order to check if these variants can bring any improvement in the sensor’s resolution for RI and magnetic field measurements. These results have been listed in Table 5.

Table 5.

Performance parameters for different variants of sensor design

Mode	Range of n4	d4 (µm)	Δ (in o)	Resolution, Rn1 (RIU) with conventional measurement	Resolution, Rn2 (RIU) with modified measurement	Remarks
RI Sensing (B = 0 T)
TM1	1–1.1	20	36.96	1.00 × 10−9	1.62 × 10−9	Rn1 and Rn2 are very close to each other in terms of their order
TM3	1–1.1	100	0.66	3.12 × 10−11	2.10 × 10−11	Rn1 and Rn2 are two orders finer than those for TM1 mode

Mode	Range of B	d4 (µm)	Δ (in o)	Resolution, RB1 (µT) with conventional measurement	Resolution, RB2 (µT) with modified measurement	Remarks
Magnetic field sensing (n4 = 1)
TM1	0–1 T	20	74	1.06	0.97	RB1 and RB2 are very close to each other
TM3	0–1 T	100	0.50	0.007	0.006	RB1 and RB2 are 2 orders finer than those for TM1 mode

The above values indicate that apart from preferring higher order of TM mode and large value of d₄, smaller values of Δ can further help in achieving even finer resolutions of both RI and magnetic field measurements. However, the above values of resolution are either comparable or inferior to those listed in previous tables. Nevertheless, the above influence of Δ on sensing performance provides another degree of freedom as far as the sensor design is concerned.

Conclusions

The results indicate that using the transverse SDS of the horizontal PSHE in a graphene- and germanium-based plasmonic sensor in THz frequency region, one may achieve very fine resolution for measurement of RI of a gas medium while magnetic field being applied to the graphene monolayer. A few salient points related to RI and magnetic field sensing are:

RI Sensing

• In the conventional weak measurements with TM₁ mode, d₄ = 20 µm, B = 0 T, and Δ = 0.1°, an RI resolution of as fine as 2.32 × 10⁻¹² RIU is achievable for gaseous medium in the range 1–1.1 RIU.

• In the modified weak measurements with TM₃ mode, d₄ = 100 µm, B = 0 T, and Δ = 0.1°, an RI resolution of 1.39 × 10⁻¹⁰ RIU is achievable for gaseous medium in the range 1–1.1 RIU.

Magnetic field sensing

• In the conventional weak measurements with TM₁ mode, d₄ = 20 µm, n₄ = 1, and Δ = 0.1°, a magnetic field resolution of 5.31 × 10⁻⁴ µT is achievable for the magnetic field in the range 0–1 T.

• In the modified weak measurements with TM₆ mode, d₄ = 200 µm, n₄ = 1, and Δ = 0.1°, a magnetic field resolution of 7.86 × 10⁻³ µT is achievable for the magnetic field in the range 0–1 T.

With larger value of d₄, the FOM of the plasmonic sensor (without PSHE measurement) increases. Also, the smaller d₄ leads to finer resolutions of RI and magnetic field with PSHE under conventional weak measurements. This trend reverses with PSHE under modified weak measurements for which larger d₄ leads to finer resolutions of RI and magnetic field (Tables 2, 3 and 4).

The amplified angle Δ can further help in achieving even finer resolutions of both RI and magnetic field particularly under modified weak measurements. (Table 5).

Some of the results discussed above are better or comparable with the existing (5 × 10⁻⁹ RIU and 0.7 µT or 1.22 × 10⁻¹¹ RIU and 1.46 × 10⁻² µT) values for a structure with four layers in the conventional weak measurement [1–3]. This study further consolidates that PSHE-based plasmonic sensor design in THz frequency can be extremely apt candidate for highly sensitive and accurate measurements of RI and magnetic field.

As far as the repeatability of this sensor design is concerned, the testing cycles of this structure should be conducted with a constant exposure and a introduction of the concerned gaseous medium (e.g., n₄ = 1.1) for few minutes. During the exposure of gases (angle θ should be fixed at particular value), the power loss magnitude will change significantly. In the next step, when the sensor is operated with air (n₄ = 1), the power loss of the sensor will shift back to its initial value. This will confirm the repeatability of the sensing device where its power loss value shift according to the RI of the gaseous medium and acquire its initial value when again its exposed with air. The similar explanation holds for variable magnetic field also.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of interest

The authors declare no conflict of interest.