Tunable asymmetric spin splitting by black phosphorus sandwiched epsilon-near-zero-metamaterial in the terahertz region

Yanmei Lin; Xiaohe Liu; Huifeng Chen; Xinyi Guo; Jintao Pan; Jianhui Yu; Huadan Zheng; Heyuan Guan; Huihui Lu; Yongchun Zhong; Yaofei Chen; Yunhan Luo; Wenguo Zhu; Zhe Chen

doi:10.1364/OE.27.015868

1. Introduction

Photonic spin splitting, refer to the spatial separation of two opposite spin components of the reflected/transmitted beam [1–4], has attracted significant attention owing to its application in precise metrology and quantum information [5–7]. The spin splitting can occur in directions both parallel and perpendicular to the plane of incidence, i.e., so-called the in-plane and out-of-plane spin splitting (IPSS and OPSS) [4–6]. Both IPSS and OPSS can be considered as results of spin-orbit coupling [1,3,4,8]. It has been demonstrated by Götte and Dennis in 2012 that the in-plane and out-of-plane spin splitting can be considered as analogous but reverse effects [9]. Different from Goos-Hänchen (GH) shift occurring in total reflection, the IPSS can appear in both cases of partial and total reflections [3,4]. In 2017, the upper limit of IPSS was derived, equal to the incident beam waist [4]. This upper limit can be obtained by optimizing the incident linear polarization state at Brewster angle [4]. Very recently, large IPSS near the critical angle of total reflection was theoretically proposed and experimentally verified [10].

Epsilon-near-zero (ENZ) metamaterial has significant interest due to its novel light-matter interactions [11]. According to the electromagnetic boundary condition, the vanishing epsilon in ENZ material will lead to strong discontinuity in the normal electric field, thus the strong field localization and enhancement [12]. The nonlinear and Purcell effects can therefore be boosted by ENZ metamaterials [13–15]. The vanishing epsilon of ENZ material also results in nearly constant phase advance when an electromagnetic field travels through the whole material [16]. Based on this phenomena, phase-mismatch-free harmonic generation, wavefront reshaping, and ultrafast phase transitions have been developed [17]. It has been demonstrated recently that ENZ metamaterials can enhance the GH shift [18] and the spin splitting [19,20].

To tune the spin splitting, we sandwich the ENZ metamaterial by monolayer black phosphorus. The light-matter interaction in this structure can be tuned by modulating the conductivity of BP, which is proportional to the electron carrier density [21]. BP possesses striking in-plane anisotropy property, enabling novel polarization-dependent and angle-resolved optoelectronic devices [22–26]. The asymmetric spin splitting induced by the in-plane anisotropy of BP was predicted very recently [27]. The reflected beam from a BP layer sitting on a silicon substrate undergoes both in-plane and out-of-plane asymmetric spin splitting near Brewster angle. However, the in-plane displacements of two opposite spin components are always positive, cannot be tuned flexibly [27]. The spin splitting of the transmitted beam from the structure are tiny small. The large asymmetric spin splitting in the transmitted beam is still missing [26,27].

Here, a novel structure combining the ENZ metamaterial and BP layers is proposed to realized large and tunable in-plane asymmetric spin splitting in the transmitted terahertz beam. The ENZ metamaterial acts as an optical cavity, resulting in an enhancement of light-matter interaction in BP layers. The BP layers can tune the asymmetric spin splitting via electron carrier density or angles between the incident plane and the armchair crystalline directions of BP layers.

2. Theory

Figure 1(a) shows the schematic of in-plane asymmetric spin splitting. An ENZ metamaterial is sandwiched by BP layers on both its top and bottom sides. The photonic property of BP layers can be described by surface conductivities. Under the Drude model, the conductivity along the armchair and zigzag crystalline directions are respectively [27,28]

σ_{a r m, z i g} = (i D_{a r m, z i g}) / [π (ω + i η / ℏ)] .

Here, D_arm,zig = πe²ρ/m_arm,zig is the Drude weight with m_arm = ћ²/(2γ²/Δ + η_c), m_zig = ћ²/2v_c being the electron mass along the armchair or zigzag directions respectively. Parameters η = 10 meV, γ = 4a/π eVm, Δ = 2eV, η_c = ћ²/0.4m₀, v_c = ћ²/1.4m₀. The scale length of BP a = 0.223 nm. The electron carrier density ρ can be changed by electric doping via a bias voltage [21–23].

Fig. 1 (a) Schematic of in-plane asymmetric spin splitting. A horizontal incident polarization can be considered as a superposition of two opposite spin, which undergo displacements X _± along x-axis thus separate spatially, after transmitted through the BP-ENZ metamaterial-BP structure. The incident plane makes angles of ϕ₁ and ϕ₂ to the armchair axes of top and bottom BP layers, respectively. (b) A single BP layer surrounding by two dielectrics with refractive index of n_j and n_j₊₁. (c) A stack of N BP layers separated by different dielectrics.

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Owing to the in-plane anisotropy of BP layer, the transmission varies when the incident plane changes with respect to the BP structure. A rotation angle ϕ is introduced, which is the angle between the incident plane and the armchair axis of BP crystal. The rotation angle can be changed practically by rotating the device with fixed incident beams. As shown by [24], the crystalline axes can be determined by analyzing the reflection of normal incident visible light with a polarizer and a CCD. The armchair and zigzag axes of BP are associated with maximum and minimum brightness of R channel, respectively [24].

For a rotation angle of ϕ, the conductance matrix connecting the surface current and electric light field can be given by $\overset{\leftrightarrow}{σ}$ = [σ_xx, σ_xy; σ_yx, σ_yy], where σ_xx = σ_armcos²ϕ + σ_zigsin²ϕ, σ_yy = σ_armsin²ϕ + σ_zigcos²ϕ, and σ_yx = σ_xy = (σ_zig-σ_arm)sinϕcosϕ [27]. The cross conductivity σ_yx results from the in-plane anisotropic of BP, which vanishes for isotropic 2D materials such as graphene.

Transfer Matrix method is a powerful tool in the analysis of light propagation through layered media [29,30]. For dielectric media, the propagation of s and p waves are described by two 2 × 2 matrices independently. However, the cross conductivity of BP will induce a coupling between p and s waves. Therefore, a 4 × 4 matrix should be employed to describe the coupling.

Consider the propagation of light across an interface formed by a BP layer that separates two dielectrics with refractive indexes n_j and n_j₊₂, as shown by Fig. 1(b). The electric field in media j and j + 1 are respectively

\begin{matrix} E_{l} = {[A_{l} k_{l z} e^{i k_{j z} z} {-B}_{l} k_{l z} e^{- i k_{l z} z}] / k_{l} {\hat{e}}_{x} + [C_{l} e^{i k_{l z} z} {-D}_{l} e^{- i k_{l z} z}] {\hat{e}}_{y} . l = j, j + 1 \\ + [- A_{l} k_{lx} e^{i k_{l z} z} {-B}_{l} k_{l x} e^{- i k_{l z} z}] / k_{l} {\hat{e}}_{z}} e^{i k_{x} z} \end{matrix}

A_l and C_l are the amplitudes of rightward waves, while B_l and D_l are of leftward waves. k_x is the x component of wavevector, which is the same in different media.

k_{l z} = {[k_{l}^{2} - k_{x}^{2}]}^{1 / 2}

with k_l being the wavenumber in is the l medium. The electromagnetic boundary conditions at the interface between media j and j + 1 are

{\hat{e}}_{z} \times [E_{j} - E_{j + 1}] = 0,

{\hat{e}}_{z} \times [H_{j} - H_{j + 1}] = \overset{\leftrightarrow}{σ} E_{j + 1},

where H_j_,_j₊₁ are the magnetic field in media j and j + 1. By substituting Eq. (2) to Eq. (3), we have

[A_{j} - B_{j}] k_{j z} / k_{j} = [A_{j + 1} - B_{j + 1}] k_{j + 1, z} / k_{j + 1},

C_{j} + D_{j} = C_{j + 1} + D_{j + 1},

[C_{j} - D_{j}] k_{j z} / ω μ = [C_{j + 1} - D_{j + 1}] k_{j + 1, z} / ω μ + σ_{y y} [C_{j + 1} + D_{j + 1}] + σ_{y x} [A_{j + 1} - B_{j + 1}] k_{j + 1, z} / k_{j + 1},

[A_{j} + B_{j}] k_{j} / ω μ = [A_{j + 1} + B_{j + 1}] k_{j + 1} / ω μ + σ_{x x} [A_{j + 1} - B_{j + 1}] k_{j + 1, z} / k_{j + 1} + σ_{y x} [C_{j + 1} + D_{j + 1}],

where ω and μ are circular frequency and permeability, respectively. Therefore, the transmission matrix for a dielectric interface containing BP layer is

T_{j, j + 1} = [\begin{matrix} k_{j + 1} / k_{j} + η_{p} + ξ_{p} σ_{x x} & k_{j + 1} / k_{j} - η_{p} - ξ_{p} σ_{x x} & ζ_{p} σ_{x y} & ζ_{p} σ_{x y} \\ k_{j + 1} / k_{j} - η_{p} + ξ_{p} σ_{x x} & k_{j + 1} / k_{j} + η_{p} + ξ_{p} σ_{x x} & ζ_{p} σ_{x y} & ζ_{p} σ_{x y} \\ ζ_{s} σ_{y x} & - ζ_{s} σ_{y x} & 1 + η_{s} + ξ_{s} σ_{y y} & 1 - η_{s} + ξ_{s} σ_{y y} \\ - ζ_{s} σ_{y x} & ζ_{s} σ_{y x} & 1 - η_{s} - ξ_{s} σ_{y y} & 1 + η_{s} - ξ_{s} σ_{y y} \end{matrix}],

where

η_{p} = k_{j} k_{j + 1, z} / k_{j + 1} k_{j z}

,

η_{s} = k_{j + 1, z} / k_{j z}

,

ξ_{p} = ω μ k_{j + 1, z} / k_{j + 1} k_{j}

,

ξ_{p} = ω μ / k_{j z}

,

ζ_{p} = ω μ / k_{j}

,

ζ_{s} = ω μ k_{j + 1, z} / k_{j z} k_{j + 1}

. If the cross conductivities σ_xy and σ_yx vanish, the upper-right and lower-left parts of the transmission matrix vanish, thus the 4 × 4 transmission matrix can be rewritten into two 2 × 2 matrices, which connect A_j, B_j with A_j₊₁, B_j₊₁ and C_j, D_j with C_j₊₁, D_j₊₁, respectively.

As the photonic property of BP layers have been described by surface conductivities, the thicknesses of BP layers are neglected. The propagation matrix in a homogenous medium j with a thickness of d_j is [29]

P_{j} = [\begin{matrix} e^{- i k_{j z} d_{j}} & 0 & 0 & 0 \\ 0 & e^{i k_{j z} d_{j}} & 0 & 0 \\ 0 & 0 & e^{- i k_{j z} d_{j}} & 0 \\ 0 & 0 & 0 & e^{i k_{j z} d_{j}} \end{matrix}] .

For the stack of N BP layers shown in Fig. 1(c), the transfer matrix can be obtained by transmission and propagation matrices. Denote the electric field coefficients on left side of the first interface by A₁, B₁, C₁, D₁ and those on the right side of the end interface by A_N+₁, B _N+₁, C_N+₁, D _N+₁. These two sets of field coefficients are then related by a 4 × 4 transfer matrix M, namely,

[\begin{matrix} A_{1} \\ B_{1} \\ C_{1} \\ D_{1} \end{matrix}] = M [\begin{matrix} A_{N + 1} \\ B_{N + 1} \\ C_{N + 1} \\ D_{N + 1} \end{matrix}],

with M = T₀₁P₁T₁₂P₂…T_N_-1,_NP _NT_N_,_N₊₁. Equation (7) is the transfer matrix to describe the light propagation through layered media with BP. For a multilayer structure without BP, both the transmission and propagation matrices can be rewritten respectively into two 2 × 2 matrices, the final matrix M can be therefore reduced into two 2 × 2 independent matrices [30].

According to Eq. (7), the Fresnel transmission coefficients can be obtained. By setting C₁ = B_{N + 1} = D_{N + 1} = 0, we have

t_{p p} = \frac{A_{N + 1}}{A_{1}} = \frac{M_{33}}{M_{11} M_{33} - M_{13} M_{31}},

t_{s p} = \frac{C_{N + 1}}{A_{1}} = \frac{- M_{31}}{M_{11} M_{33} - M_{13} M_{31}},

Similarly, by setting A₁ = B_{N + 1} = D_{N + 1} = 0, we have

t_{s s} = \frac{C_{N + 1}}{C_{1}} = \frac{M_{11}}{M_{11} M_{33} - M_{13} M_{31}},

t_{p s} = \frac{A_{N + 1}}{C_{1}} = \frac{- M_{13}}{M_{11} M_{33} - M_{13} M_{31}},

In our case, two BP layers are considered. The electron carrier densities and rotation angles (Angle between the armchair axis of BP crystal and the x_g-axis) of top and bottom BP layers are respectively ρ₁, ϕ₁ and ρ₂, ϕ₂, as shown in Fig. 1(a). The carrier densities are set to be the same, i.e., ρ₁ = ρ₂ = ρ, through the whole paper.

Considering a horizontally (H) polarized Gaussian incident beam tunneling through the BP multi-layer structure. The spectrum of incident beam is ${\tilde{E}}_{i} = \exp [- (k_{x}^{2} + k_{y}^{2}) w_{0}^{2} / 4] | H 〉$ where w₀ being the beam waist. According to [31,32], the transmitted spectrum can be given by ${\tilde{E}}_{t} = Q {\tilde{E}}_{i}$ . The transformation matrix Q can be expressed as

Q = [\begin{matrix} t_{p p} + κ_{x} t_{p p}^{'} - κ_{y} Ι & t_{p s} + κ_{x} t_{p s}^{'} + κ_{y} Κ \\ t_{s p} + κ_{x} t_{s p}^{'} + κ_{y} Κ & t_{s s} + κ_{x} t_{s s}^{'} + κ_{y} Ι \end{matrix}],

for the case with the same material in the first and last media. I = (t_sp + t_ps)cotθ, K = (t_pp-t_ss)cotθ,

t_{i j}^{'}

are the first derivate of t_ij. κ_x_,_y = k_x,_y/k₀ with k₀ being the wavenumber in vacuum. In the circular polarization basis, the right- and left-handed circular polarization (RCP and LCP) components of the transmitted beams for H incident polarization are:

{\tilde{E}}_{t}^{\pm} = [(t_{p p} + κ_{x} t_{p p}^{'} - κ_{y} Ι) \mp i (t_{s p} + κ_{x} t_{s p}^{'} + κ_{y} Κ)] {\tilde{u}}_{0} | \pm 〉

The RCP and LCP components are no longer maintaining the Gaussian envelope, and their centroids may shift along x-axis. With respect to the geometric prediction, the centroid displacements can be defined as

X_{\pm} = \iint {\tilde{E}}_{t}^{\pm} \partial_{k_{r x}} {\tilde{E}}_{t}^{\pm} | d k_{x} d k_{y} / \iint | {\tilde{E}}_{t}^{\pm} |^{2} d k_{x} d k_{y}

[32]. After some straightforward calculation, we obtain

X_{\pm} = {- Im [t_{p p}^{*} t_{p p}^{'} + t_{s p}^{*} t_{s p}^{'}] \pm Re [t_{p p}^{*} t_{s p}^{*} - t_{s p}^{*} t_{p p}^{'}]} / k_{0} W_{\pm},

where the energies of RCP and LCP components are

\begin{array}{l} W_{\pm} = | t_{p p} |^{2} + | t_{s p} |^{2} \pm 2 Im | t_{p p}^{*} t_{s p} | + \frac{1}{k_{0}^{2} w_{0}^{2}} {| t_{p p}^{'} |^{2} + | t_{s p}^{'} |^{2} \\ \pm 2 Im | t_{p p}^{'}^{*} t_{s p}^{'} | + | Κ |^{2} + | Ι |^{2} \mp 2 Im [Κ^{*} Ι]} \end{array}

The first term of Eq. (11) is the GH shift, moving the RCP and LCP components together. The second term is spin-dependent, shifting the two opposite spin components toward opposite directions. This spin dependent term vanishes for a vanishing t_sp. Generally, the cooperation effect of the first and second terms will induce an asymmetric spin splitting.

3. Results and discussion

Firstly, we consider the case with the rotation angles of top and bottom BP layers being ϕ₁ = ϕ₂ = 0. Thus, the cross conductivities are zero, resulting in the vanishing of the spin-dependent term in Eq. (11). Therefore, the transmitted beam undergoes pure GH shift without spin splitting. Figure 2(a) shows the GH shifts changing with the incident angle θ for different carrier density ρ when the refractive index and thickness of ENZ metamaterial are n = 0.1 and d = 0.1 mm, respectively. Here, we focus our attention on the terahertz region by setting the frequency incident beam being 1 THz, since this region is underdeveloped but ripe for exploitation [33]. For each carrier density, the GH shift reaches a peak at θ = 6.12°. The peak GH shift increases with the carrier density, which is clearly shown by the blue line in Fig. 2(c). Thus, the GH shift can be enhanced by the BP layers, since ρ = 0 corresponds to the case without BP. At other incident angle such as θ = 3.5°, the GH shift may decreases with the carrier density.

Fig. 2 (a) GH shifts changing with the incident angle for the carrier density ρ = 0, 2.5, and 5 × 10¹⁷ m⁻², and for (b) the refractive index of ENZ metamaterial n = 0.1, 0.2. 0.4, 0.6, respectively. (c,d) GH shifts changing with the carrier density ρ (c) and the metamaterial thickness d (d). In the calculation, the frequency of the incident beam is 1 THz.

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The GH shift depends on the refractive index n and thickness d of the ENZ metamaterial. As shown by Fig. 2(b), with the increase of n, the peak value of GH shift decreases and the peak position moves, indicating that a larger shift can be obtain by a smaller refractive index of the metamaterial. Figure 2(d) gives the GH shift increases monotonously with thickness d, when ρ = 5 × 10¹⁷m⁻², θ = 6.12°.

By rotating the BP layer, the transmitted beam undergoes asymmetric spin splitting. The displacements of two opposite spin components of the transmitted beam are given in Fig. 3(a) as function of the incident angle when ϕ₁ = ϕ₂ = 45°, d = 4.2 μm, n = 0.1. In this special case, the displacements of RCP component X₊ are almost vanish for all carrier densities. However, the displacements of LCP component X_- can take large values. When the carrier density increases from 2 × 10¹⁷ to 5 × 10¹⁷m⁻², the peak displacement decreases gradually, and change its sign at ρ = 4.7 × 10¹⁷m⁻². Meanwhile, the peak position moves from 71.5° to 50.5° gradually. At the peak positions, owing to the strong light-interaction in BP layers, the transmission coefficients |t_ps| are almost identical with coefficients |t_pp| (see Fig. 3(b)). And they are with nearly −90° phase difference (φ = arg[t_pp/|t_ps]), as shown by Fig. 3(b). Therefore, the first three terms of W_- in Eq. (12) almost vanishes, leading to large displacement in X_-. The phenomenon was also found at air-chiral metamaterial interface [34].

Fig. 3 Displacements of two opposite spin components of the transmitted beam X₊ (solid lines) and X_- (dotted lines) (a), transmission coefficients |t_pp| and |t_ps| (b), and phase difference φ = arg[t_pp/|t_ps] (c) changing with the incident angle, respectively. In the calculation, d = 4.2 μm, n = 0.1, ϕ₁ = ϕ₂ = 45°.

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The refractive index and the thickness of the ENZ metamaterial affect the asymmetric spin splitting in the transmitted beam. Figures 4(a) and 4(b) show the displacements X _± changing with the incident angle θ for the refractive index n = 0.05, 0.1. 0.2 (a), and for n = 0.1, 0.1 + 0.01i, 0.1 + 0.1i(b), respectively. One can find from Fig. 4(a) that, the maximum value of displacements |X _± | as well as the maximum displacement difference |X₊-X_-| decreases with the increase of the real part of the refractive index Re[n]. Therefore, a near-zero permittivity can enhance the asymmetric spin splitting. The near-zero permittivity can be achieved by both artificial and natural materials. Polar dielectrics like LiF, NaCl, and GaAs are the natural terahertz ENZ materials with the real part of permittivity close to zero near the polaritonic resonance frequency [35]. Effective terahertz ENZ materials have been realized by rectangular waveguide [36], graphene-dielectric composite structure [37], etc.

Fig. 4 Displacements of two opposite spin components X _± changing with the incident angle θ for the refractive index of ENZ metamaterial n = 0.05, 0.1. 0.2 (a), and for n = 0.1, 0.1 + 0.01i, 0.1 + 0.1i (b), respectively. (c) X _± changing with the metamaterial thickness d for different θ. In the calculation, ϕ₁ = ϕ₂ = 45°.

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The loss of the metamaterial can be described by the image part of the refractive index Im[n] [18]. As shown by Fig. 4(a), Im[n] will reduce the displacements X _± . However, for the Im[n] smaller than 0.01, the displacements X _± are almost unchanged. Figure 4(c) shows the displacements X _± changing with the metamaterial thickness d when n = 0.1. The displacements X _± will approach to asymptotic values when the thickness d increases. For a larger incident angle, both X₊ and X_- possess a sharper peak.

The asymmetric spin splitting in the transmitted beam depends strongly on the rotation angle of BP layers, as shown in Fig. 5. In Fig. 5, the rotation angles of the top and bottom layers are equal, i.e., ϕ = ϕ₁ = ϕ₂. The displacements X₊ and X_- are mirror symmetric about ϕ = 90°. The displacements of the RCP component X₊ takes large values ϕ>90°, while X_- for ϕ<90° . When θ = 44°, X₊ has a positive peak, and become two positive peaks for θ = 47°. When θ>50°, a positive and a negative peaks can be found in the pattern of X₊. And distance between this two peaks enlarges with the increase of the incident angle θ.

Fig. 5 Displacements X _± as functions of the rotation angle ϕ, where ϕ = ϕ₁ = ϕ₂·. The other parameters are the same as Fig. 3.

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In the above case, one spin component has large displacement while the other almost vanishes. Figure 6(a) gives the displacements X _± changing with the incident angle for the thickness d = 0.15 mm the carrier density ρ = 0 and 5 × 10¹⁷m⁻², respectively. Without BP layers (ρ = 0), X₊ and X_- coincide each other. When ρ = 5 × 10¹⁷m⁻², however, X₊ and X_- are different excepting for ϕ = 0, 90°, or 180°. At these three angles, the two opposite spin components of the transmitted beam do not split. However, the two spin components may split if the incident beam is elliptically polarized [4]. At θ = 6.21°, both X₊ and X_- are positive. However, X₊ and X_- are opposite in sign at θ = 11.5°. Figures 6(b) and 6(c) show the displacements X _± changing with the rotation angle for θ = 6.21° and 11.5°, respectively. The displacements vary with rotation angle flexibly.

Fig. 6 Displacements X _± as functions of the incident angle θ for ρ = 0 and 5 × 10¹⁷m⁻², when d = 0.15 mm, ϕ₁ = ϕ₂ = 45° (a). X _± changing with the rotation angle ϕ (ϕ = ϕ₁ = ϕ₂) for θ = 6.21° (b) and 11.50° (c), respectively. The inset in (b) shows the rotation of BP-metamaterial structure. Schematics of barcode encryption based on the asymmetric spin splitting are also shown by choosing the threshold as 0 (b) and 1.8 mm (c), respectively.

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The asymmetric displacements for LCP and RCP components of the transmitted beam along x-axis can be regarded as two independent channels for information processing [27]. The centroid displacements larger than a certain threshold value can be considered as “1”, while smaller than that value as “0”. The encoding rule varies with the choice of the threshold values. For the case of θ = 11.5°, three different codes: “11”, “10”, “01” can be obtained by tuning the rotation angle with a threshold of 0. All four different codes: “11”, “10”, “01”, “00” are found for the case of θ = 6.21° with a threshold of 1.8 mm. This is comparing to best three different codes in [27] and [38].

Twisted bilayer graphene have attracted much attention recently owing to its unconventional superconductivity at the magic-angle [39]. When the rotation angles of the top and bottom BP layers are different, the transmitted beam shows interesting spin splitting phenomena. Figure 7 shows the displacements X _± as functions of ϕ₁ and ϕ₂ when θ = 14.14° and d = 0.15 mm. X₊ and X_- are almost symmetric about the centroid point of (ϕ₁,ϕ₂) = (90°,90°). Both X₊ and X_- will change sign when ϕ₂ crosses 90°. With the change of ϕ₁, only the magnitude of X _± will change.

Fig. 7 Displacements X₊ (a) and X_- (b) as functions the rotation angles ϕ₁ and ϕ₂ when θ = 14.14°, d = 0.15 mm, ρ = 5 × 10¹⁷m⁻².

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Figure 8 shows Displacements X _± changing with the rotation angle ϕ₂ for different ϕ₁ when θ = 10° (a), 20° (b),30° (c), respectively. When θ = 10°, there is one peaks for X _± in the ϕ₂ region of 0-90°. When θ = 20° and 30°, the X_- become two peaks in the same region, while the X₊ are nearly flat.

Fig. 8 Displacements X₊ (solid lines) and X_- (dotted lines) changing with the rotation angle ϕ₂ for different ϕ₁ and θ, when d = 0.05 mm, ρ = 5 × 10¹⁷m⁻².

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The ENZ metamaterials sandwiched with BP layers show unique photonic spin splitting effect. Although the CVD growth of 2D black phosphorus film has been demonstrated in 2016, the large-scale of monolayer BP is still challenge [40]. Hopeful methods have been proposed recently to address this challenge [41]. The unique photonic spin splitting effect based on BP can be readily extended to other anisotropic 2D materials such as ReSe₂ and ReS₂, for which high quality large-scale atomic films are well reachable [42].

4. Conclusions

In conclusion, we have extended the transformation matrix to describe the propagation in a multi-layer dielectric stack containing BP layers. A novel air-BP-metamaterial-BP-air structure is proposed to enhance the in-plane photonic spin splitting effect. The transmitted beam through the structure undergoes asymmetric spin splitting due to the strong in-plane anisotropy of BP layers. The asymmetric splitting depends strongly on the rotation angles of BP layers, namely, ϕ₁ and ϕ_2. However, only the GH shift occurs without spin splitting for ϕ₁ = ϕ₂ = 0 or 90°. Based on the tunable asymmetric spin splitting, a two-channel angle-resolved barcode encoding is achieved. The photonic spin splitting effect in BP layers promises novel angle-resolved optoelectronic devices in the Terahertz region.

Funding

National Natural Science Foundation of China (61705086, 61675092, 61475066); Natural Science Foundation of Guangdong Province (2017A030313375, 2016TQ03X962, 2017A010102006, 2016A030311019, 2016A030313079, 2017A030313359); Science & Technology Project of Guangzhou (201803020023, 201707010396, 201704030105, 201605030002, 201604040005).

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Tunable asymmetric spin splitting by black phosphorus sandwiched epsilon-near-zero-metamaterial in the terahertz region

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusions

Funding

References

Cited By

Figures (8)

Equations (19)

Optics Express