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Abstract
A novel terahertz (THz) sensing scheme is proposed based on the photonic spin Hall effect (PSHE). By illumining a paraxial Gaussian THz beam onto a black phosphorus (BP)-based Tamm structure, the reflected beam will undergo in-plane spin splitting, i.e., the centroids of two opposite spin components separate spatially. Due to Tamm plasmon resonance, one of the spin components is very sensitive to the refractive index changes of the analyte layer sandwiched by monolayer BP and distributed Bragg reflector. The sensitivity of the spin-dependent shift can be up to 2804 mm/RIU with a refractive index resolution of ∼10−8 RIU. The sensitivity and dynamic sensing region can be flexibly tuned by the BP rotation angle, thickness of analyte layer, or operation frequency. Therefore, the proposed PSHE-based THz sensing provides a new avenue for the development of high-performance THz sensors; thus, we may find applications in chemical sensing and biosensing.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The terahertz (THz) wave, lying between 0.1 and 10 THz, connects the microwave and the infrared optical wave [1,2]. The low photon energy of the THz wave will not induce radiative ionization damage for biological molecules. The THz waves mainly excite the intramolecular and intermolecular vibrations, thus the THz absorption spectrum are considered as distinctive spectral signatures of most molecules and biomolecules [2]. Recently, much attention has been focused on the THz sensing of biological materials. To enhance the interaction between molecules and THz wave, plasmonic resonances are frequently employed [3,4]. As a band gapless two-dimensional (2D) material, graphene can support THz surface plasmons excellently [5,6]. Different graphene-combined structure have been demonstrated for enhanced THz sensing, such as multilayer structure [7], attenuated total reflection configure [8,9], three-dimensional graphene metamaterial structure [10], nano-slot-based resonance structure [11], etc. Black phosphorus (BP) is another promising 2D material in THz applications [12–16]. The unique in-plane anisotropic and tunable bandgap properties enable BP to develop polarization-sensitive devices [17–19]. Although many THz photodetectors [14] and perfect absorbers [20] have been demonstrated, the THz sensing application based on BP is still missing.
On the other hand, photonic spin Hall effect (PSHE) refers to photons with opposite spin split spatially when reflected from or transmitted through an interface between two medium, namely, the intensity profiles of the right and left circular polarization (RCP and LCP) components of reflected/transmitted beam have a relative movement [21–26]. The spin-dependent shifts resulted from PSHE are very sensitive to the refractive indexes of the medium, thus have been used in optical sensing. Based on the PSHE, Luo group not only identified the layer number of graphene [27], but also measured its conductivity [28]. In 2016, Qiu et al. measured the weak optical rotation induced by chiral molecules via PSHE, thus estimated the concentration of glucose and fructose [29]. Based on this fact, Wang et al. real-time monitored the sucrose hydrolysis [30]. The optical rotation changes during the sucrose hydrolysis process, which is detected precisely by measuring the spin-dependent shifts [30].
Here, we propose a BP-based THz sensor based on the PSHE. The in-plane anisotropic of the BP leads to asymmetric spin splitting, where the displacements of two opposite spin components are different. A one-dimensional waveguide structure with a monolayer BP on the top is designed to excite the Tamm plasmons, and the strong light-matter interaction in Tamm plasmon resonance enhances the asymmetric spin splitting. We find that one of the spin-dependent shift is very sensitive to the refractive index changes of the analyte layer. The sensing region can be well controlled by the thickness of the analyte layer or operated frequency.
2. Model and theory
As shown in Fig. 1, the BP-based Tamm structure consists of a monolayer of BP, an analyte layer, and a distributed Bragg reflector (DBR). The DBR is composed of 15 alternating dielectric layers of silicon (Si) and silicon dioxide (SiO2). The refractive indices of Si and SiO2 are respectively 3.42 and 1.9 with quarter-wave layer thickness for a center frequency of 1 THz, namely, dSi=21.93 μm, dSiO2=39.47 μm. BP shows metallic property in the THz region, acting as the top reflected layer to form an asymmetric cavity. In the cavity, the analyte layer with a thickness of ds and a refractive index of ns is sandwiched by the BP and DBR. Thus, the strong interaction between light and matter occurs in analyte layer, which enhances the analyte sensing performance. A glass prism is used to couple the incident beam to Tamm plasmon and as a supporter for BP layer.
Fig. 1. Schematic of BP THz sensor based on photonic spin Hall effect. A H-polarized Gaussian beam reflected by a BP-based Tamm structure splits into two opposite spin components with a spatial displacement along the x axis.
Let us consider a monochromatic beam illuminating onto BP-based Tamm structure. The photonic property of BP can be described by a surface conductivity [31]. Under the Drude model, it is [31]
where Darm,zig=πe2n/marm,zig is the Drude weight. marm=ћ2/(2γ2/Δ+ηc), mzig=ћ2/2vc are respectively the electron mass along the armchair and zigzag directions, where η=10 meV, Δ=2 eV, ηc=ћ2/0.4m0, vc= ћ2/1.4m0, γ=4a/π eVm, with the scale length of BP being a=0.223 nm. The electron carrier density ρ can be changed by electric doping via a bias voltage. σarm,zig is the conductivity along the armchair or zigzag crystalline directions.By rotating the BP, the armchair axis of BP crystal makes an angle of ϕ with the incident plane. Therefore, the conductance matrix connecting the surface current and electric light field can be given by σ=[σpp, σps; σsp, σss], where σpp=σarmcos2ϕ+σzigsin2ϕ, σss=σarmsin2ϕ+σzigcos2ϕ, and σsp=σps=(σzig-σarm)sinϕcosϕ [15].
To describe the light propagation through layered medium containing with BP layers, a modified transfer matrix has been derived very recently [32]. A 4×4 matrix should be employed since the cross conductivity of BP σsp will induce a coupling between p and s waves. As illustrated by Ref. [32], the 4×4 transfer matrix is $M = T_{0,s}^{BP}{P_s}{T_{a,b}}{P_b}{T_{b,a}}{P_a}\ldots {T_{b,a}}{P_a}{T_{b,a}}$, where $T_{0,s}^{BP}$ describes an air-analyte layer interface containing BP layer, while ${T_{a,b}}$ and ${T_{b,a}}$ describe the dielectric a-b and dielectric b-a interface, respectively. Ps, Pa, Pb are the propagation matrix in analyte layer and medium a and b, respectively. The Fresnel reflection coefficients can be therefore calculated as:
Considering a horizontally (H) polarized Gaussian incident beam reflected by the BP-based Tamm structure. The spectrum of incident beam is ${\tilde{{\textbf E}}_i} = exp [{ - (k_x^2 + k_y^2)w_0^2/4} ]|H \rangle$ where w0 being the beam waist. According to [15,16], the reflected spectrum is connected with the incident one, and can be given by ${\tilde{{\textbf E}}_r} = {\textbf Q}{\tilde{{\textbf E}}_i}$. The transformation matrix Q is given by
3. Results and discussions
According to Eqs. (2a)–(2d), the Fresnel reflection coefficients |rss|, |rpp|, and |rps| are calculated at normal incidence. The numerical results are shown in Fig. 2. The |rpp| has a dip down to zero at frequency f=1 THz when the thickness and refractive index of the analyte layer are ds=150 μm and ns=1.35, respectively. However, the dip of |rss| undergoes a redshift and can only go down to 0.41. The vanishing |rps| is not shown as the rotation angle is ϕ=0. As illustrated by Fig. 2(c), the dip position |rpp| decreases linearly with the increase of the refractive index of analyte ns with a slope of -0.35 THz/RIU. Therefore, the proposed BP-based Tamm structure can be used as a THz spectrum sensor with high sensitivity and large sensing range (>0.2 RIU).
Fig. 2. Reflection coefficients |rss| (solid lines), |rpp| (doted lines), and |rps| (dot-dashed line) as functions of frequency f at normal incidence for different refractive index ns (a), different carrier density ρ (b), and different rotation angle ϕ. (c) The dip frequency changing the refractive index ns.
Figure 2(b) shows the Fresnel reflection coefficients changing with frequency for different carrier density ρ. Without BP (ρ=0), the |rss| and |rpp| are identical and overlapped with a stop band centered at 1 THz with a bandwidth of ∼0.36 THz due to the DBR. With the increase of carrier density, the in-plane anisotropy of BP breaks the identical property. When ρ=7.2×1017 m-2, |rpp| reaches a zero dip at f=1 THz owing to the excitation of Tamm plasmon [32]. As shown by Fig. 2(d), when rotates the BP, |rps| arises. The |rps| reaches maximum value when rotation angle is ϕ=45o.
In the following, we focus on the PSHE of BP-based Tamm structure. The spin-dependent shifts Δ± changing with the rotation angle ϕ are shown in Fig. 3(a) when the incident angle is θ=52.4o. The shifts of two opposite spin components of reflected beam are identical at ϕ=0, 90o, and 180o. Generally, however, they are different, i.e. the two photonic spin split asymmetrically. This asymmetric spin splitting phenomenon will not occur in isotropic 2D materials such as graphene. The shift of left-handed circular component |Δ-| is larger than that of right-handed one |Δ+| in the region of 0<ϕ<90o. For the region of 90o <ϕ<180o, |Δ+| is larger than |Δ-|. The spin component with larger displacements has a lower energy, as illustrated by Fig. 3(b). This is because the energies happen to be the denominators of the spin-dependent shifts, as shown by Eq. (5).
Fig. 3. Displacements Δ± (a) and energy W± (b) of two spin components of the reflected beam changing with the rotation angle ϕ when the incident angle is θ=52.4o, the thickness and refractive index of the analyte layer are ds=150 μm and ns=1.34, respectively.
The spin-dependent shifts are verified by numerical simulation with commercial software Comsol Multiphysics. To simply the simulation, a 2D model is used, meaning that the in-plane spin-dependent shifts for a 1D horizontally polarized Gaussian beam is simulated. As shown by Fig. 4(a), 1 THz Gaussian beam with waist radii of 3 mm illuminates obliquely the BP-based Tamm structure with an incident angle of 30.5o. The period of the Si-SiO2 Bragg structure is 10. The BP layer is represented by surface conductivity, introduced to model by setting surface current at the boundary between prism and analyte. The output port is perpendicular to the reflected beam; thus, we can get the reflected beam in its own coordinate directly. The output port and bottom substrate are surrounded with perfect matched layers of 0.6 mm thickness to absorb the outgoing waves; the remaining boundaries are set as scattering boundaries to make the boundary transparent, as shown by Fig. 4(a).
Fig. 4. (a) Diagonal of Comsol simulation model, where the inset shows the structure of distributed Bragg reflector. (b) The simulated intensity distribution of total electric light field with a 1 THz beam incident. (c) The normalized intensity profiles of the reflected RCP and LCP components I±. The intensity of LCP component I- has been multiplied to 50. The centroid shifts of RCP and LCP components are 0.09 and 0.8 mm, respectively.
The simulated results are shown in Fig. 4(b). Only 7.5 percent of the incident energy is reflected to the output port, as most light is absorbed by the BP layer. The shift of the whole beam, i.e., the GH shift is ΔGH=0.09 mm, thus the displacement of the reflected beam is not obvious. When inspecting the two opposite spin components of the reflected beam, we will find different scenarios. The shift of the RCP component is Δ+=0.083 mm, almost identical to the GH shift. However, the shift of the LCP component is up to 0.82 mm. The movement of intensity profile of LCP component is obvious, as clearly shown by Fig. 4(c). The theoretical shift of the LCP component is predicted according to Eq. (5), which is 0.80 mm, in good agreement with the simulated results. It is worth to note that the energy of the LCP component is ∼80 times smaller than that of the RCP one.
Further, we compare the spin-dependent shifts |Δ±| and GH shift |ΔGH|. As shown by Fig. 5, these shifts vary with the carrier density and the rotation angle of BP. The shifts will change their signs when the carrier density increases, thus a negative peak and a positive peak can be found. For the spin-dependent shifts |Δ+|, the peak is around point (ϕ,ρ)=(4.1,7.2×1017), which move to (ϕ,ρ)=(-4.1,7.2×1017) for |Δ-|. The maximum values of the spin-dependent shifts |Δ±| are identically equal to 4.5 mm. In contrast, the maximum value of the GH shift |ΔGH| is only 0.51 mm, which is ∼9 times smaller than the spin-dependent shifts. Besides, the maximum GH shift is obtained at ϕ=0. Therefore, one can expect that the spin-dependent shifts would have better performance than the GH shift in optical sensing applications.
Fig. 5. Spin-dependent Δ± (a, b) and GH shifts ΔGH (c) as functions of rotation angle and carrier density when the incident angle is θ=25o.
The spin-dependent shifts are very sensitive to the refractive index changes of the analyte layer. Figure 6(a) shows the spin-dependent shifts Δ± changing with the refractive index ns for the rotation angle ϕ=3o, 3.3o, 3.6o, 3.9o, and 4.2o, respectively. With the increase of the rotation angle, the maximum values of |Δ±| increase gradually with their position move toward the big-end gradually. The maximum values of |Δ-| are always larger than those of |Δ+|. When the thickness of analyte layer varies, the positions of the maximum values also move, as shown by Fig. 6(b).
Fig. 6. (a,b) Spin-dependent shifts Δ± changing with refractive index for rotation angle ϕ=3o, 3.3o, 3.6o, 3.9o, and 4.2o; and for thickness of analyte layer ds=142, 145, 148, 151, and 154 μm, respectively. (c,d) The corresponding maximum sensitivity of Δ± for different ϕ (c) and different ds (d), respectively.
The sensitivities of the spin-dependent shifts to refractive index are defined as S±=dΔ±/dns. The maximum sensitivities of two opposite spin are shown in Figs. 6(c) and (d) for different rotation angle and thickness. As shown by Fig. 6(c), when the rotation angle increases from 3o to 4.2o, the sensing regions for the right-handed circular and left-handed circular components are different, which are 1.34∼1.355 and 1.368∼1.373, respectively. The sensitivity of S- is up to 2804 mm/RIU, much larger S+. Therefore, the shift of the reflected LCP component is more suitable for the sensing application. At the maximum sensitivity S- = 2804 mm/RIU, the normalized energy of LCP component is W-=4×10−5. When the thickness increases from 142 to 154 μm, the maximum sensitivity of S- has a small decrease from 2804 to 2206 mm/RIU. And the position for the maximum sensitivity moves from 1.425 to 1.344 gradually. The submillimeter thickness of the analyte layer allows us to fabricate a fluid channel between the DBR and BP/prism, thus a compact BP-based sensing device.
Finally, the influence of the frequency on the spin-dependent shifts are investigated. As shown by Fig. 7, the increase of the operation frequency moves the peaks of the spin-dependent shifts. The peak of Δ+ moves from 1.391 to 1.327, while that of Δ- moves from 1.367 to 1.304, when the frequency increases from 0.99 to 1.02 THz. Therefore, one can conclude that, the sensitivity and sensing region can be flexibly tuned by the BP rotation angle, thickness of analyte layer, and operation frequency.
The resolution of refractive index is an important factor in optical sensing. Our THz sensor based on the spin-dependent shift relies on the spatial resolution of the THz imager. There are many different kinds of THz imaging methods have been developed, such as THz field-effect transistor-based camera, THz thermal camera, THz imaging with single-pixel detector, and THz near-field imaging [35]. Among them, then near-field imaging has highest spatial resolution. In 2018, Zhang et al. imaged the exfoliated single and multilayer graphene flakes by using a scattering type near-field optical microscope (s-SNOM) with spatial resolution of <100 nm [36]. Therefore, the minimum refractive index resolution can be down to 3.66×10−8 RIU.
4. Conclusions
A novel THz sensor has been proposed based on the PSHE in BP-based Tamm structure. The in-plane anisotropy of BP induced polarization-sensitive Tamm plasmon, resulting in asymmetric spin splitting. One of the spin components of the reflected beam undergoes giant shift of about 9 times larger than the GH shift. The spin-dependent shifts are sensitive to the refractive index change in analyte. The sensitivity can reach 2804 mm/RIU with an expected refractive index resolution of ∼10−8 RIU. Although the PSHE-based THz sensor with a relatively small dynamic sensing range, the sensing region can be flexibly controlled by the BP rotation angle, thickness of analyte layer, and operation frequency. Therefore, a high sensitivity and large dynamic sensing region can be obtained, which is highly desirable for developing high performance THz sensor.
Funding
National Natural Science Foundation of China (61475066, 61505069, 61705086); Natural Science Foundation of Guangdong Province (2016A030311019, 2016A030313079, 2016TQ03X962, 2017A010102006, 2017A030313359, 2017A030313375); H2020 Marie Skłodowska-Curie Actions (872172); College Students’ Innovative Entrepreneurial Training Plan Program (201910559054).
Disclosures
The authors declare no conflicts of interest.
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