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Journals >Photonics Research >Volume 6 >Issue 6 >Page 511 > Article

Photonics Research
Vol. 6, Issue 6, 511 (2018)

Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals

Wenshuai Zhang¹, Weijie Wu¹, Shizhen Chen¹, Jin Zhang¹, Xiaohui Ling¹, Weixing Shu², Hailu Luo^1、2、*, and Shuangchun Wen²

Author Affiliations

¹Laboratory for Spin Photonics, School of Physics and Electronics, Hunan University, Changsha 410082, China

²Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, School of Physics and Electronics, Hunan University, Changsha 410082, China

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DOI: 10.1364/PRJ.6.000511 Cite this Article Set citation alerts

Wenshuai Zhang, Weijie Wu, Shizhen Chen, Jin Zhang, Xiaohui Ling, Weixing Shu, Hailu Luo, Shuangchun Wen. Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals[J]. Photonics Research, 2018, 6(6): 511 Copy Citation Text

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Abstract

We examine the spin-orbit interaction of light and photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals. As an example, the photonic spin Hall effect on the surface of black phosphorus is investigated. The photonic spin Hall effect manifests itself as the spin-dependent beam shifts in both transverse and in-plane directions. We demonstrate that the spin-dependent shifts are sensitive to the orientation of the optical axis, doping concentration, and interband transitions. These results can be extensively extended to other anisotropic two-dimensional atomic crystals. By incorporating the quantum weak measurement techniques, the photonic spin Hall effect holds great promise for detecting the parameters of anisotropic two-dimensional atomic crystals.

Keywords

Electromagnetic optics Polarization

σ_{i i} = \frac{I e^{2}}{ω + I η} \frac{n}{m_{i}} + s_{i} \frac{e^{2}}{4 ℏ} [Θ (ω - ω_{i}) + \frac{I}{π} \ln | \frac{ω - ω_{i}}{ω + ω_{i}} |],

(1)

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M_{0} = (\begin{matrix} σ_{x x} & 0 \\ 0 & σ_{y y} \end{matrix}),

(2)

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M = M_{l} M_{0} M_{r} .

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M_{l} = (\begin{matrix} \cos ϕ & \sin (- ϕ) \\ \sin ϕ & \cos ϕ \end{matrix}),

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M_{r} = (\begin{matrix} \cos ϕ & \sin ϕ \\ \sin (- ϕ) & \cos ϕ \end{matrix}),

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σ_{p p} = σ_{x x} \cos^{2} ϕ + σ_{y y} \sin^{2} ϕ,

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σ_{p s} = σ_{s p} = (σ_{x x} - σ_{y y}) \cos ϕ \sin ϕ,

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σ_{s s} = σ_{y y} \cos^{2} ϕ + σ_{x x} \sin^{2} ϕ .

(8)

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E_{i}^{s} + E_{r}^{s} = E_{t}^{s},

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\cos θ_{i} (E_{i}^{p} - E_{r}^{p}) = \cos θ_{r} E_{t}^{p},

(10)

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\frac{\cos θ_{i}}{Z_{0}} (E_{i}^{s} - E_{r}^{s}) = (σ_{s s} + \frac{\cos θ_{r}}{Z}) E_{t}^{s} + σ_{s p} \cos θ_{r} E_{t}^{p},

(11)

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\frac{1}{Z_{0}} (E_{i}^{p} + E_{r}^{p}) = (σ_{p p} \cos θ_{r} + \frac{1}{Z}) E_{t}^{p} - σ_{p s} E_{t}^{s} .

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r_{p p} = \frac{α_{+}^{T} α_{-}^{L} + β}{α_{+}^{T} α_{+}^{L} + β},

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r_{s s} = - \frac{α_{-}^{T} α_{+}^{L} + β}{α_{+}^{T} α_{+}^{L} + β},

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r_{p s} = - r_{s p} = \frac{γ}{α_{+}^{T} α_{+}^{L} + β} .

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| H (k_{i, r}) ⟩ = | P (k_{i, r}) ⟩ - \frac{k_{i y}}{k_{i, r}} \cot θ_{i, r} | S (k_{i, r}) ⟩,

(16)

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| V (k_{i, r}) ⟩ = | S (k_{i, r}) ⟩ + \frac{k_{i y}}{k_{i, r}} \cot θ_{i, r} | P (k_{i, r}) ⟩,

(17)

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m_{R} = (\begin{matrix} r_{p p} & r_{p s} \\ r_{s p} & r_{s s} \end{matrix}) .

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| H (k_{i}) ⟩ \to [r_{p p} - \frac{k_{r y} \cot θ (r_{p s} - r_{s p})}{k_{0}}] | H (k_{r}) ⟩ + [r_{s p} - \frac{k_{r y} \cot θ (r_{p p} + r_{s s})}{k_{0}}] | V (k_{r}) ⟩,

(19)

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| V (k_{i}) ⟩ \to [r_{p s} + \frac{k_{r y} \cot θ (r_{p p} + r_{s s})}{k_{0}}] | H (k_{r}) ⟩ + (r_{s s} - \frac{2 k_{r y} \cot θ r_{p s}}{k_{0}}) | V (k_{r}) ⟩,

(20)

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| H ⟩ = \frac{1}{\sqrt{2}} (| + ⟩ + | - ⟩),

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| V ⟩ = \frac{1}{\sqrt{2}} i (| - ⟩ - | + ⟩),

(22)

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| ϕ ⟩ = \frac{w_{0}}{\sqrt{2 π}} \exp [- \frac{w_{0}^{2} (k_{i x}^{2} + k_{i y}^{2})}{4}],

(23)

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| ϕ_{r}^{H} ⟩ \approx \frac{r_{p p} + i r_{p s}}{\sqrt{2}} \exp (+ i k_{r x} δ_{x}^{H} + i k_{r y} δ_{y}^{H}) | + ⟩ | ϕ ⟩ + \frac{r_{p p} - i r_{s p}}{\sqrt{2}} \exp (- i k_{r x} δ_{x}^{H} - i k_{r y} δ_{y}^{H}) | - ⟩ | ϕ ⟩,

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| ϕ_{r}^{V} ⟩ \approx \frac{r_{p s} - i r_{s s}}{\sqrt{2}} \exp (+ i k_{r x} δ_{x}^{V} + i k_{r y} δ_{y}^{V}) | + ⟩ | ϕ ⟩ + \frac{r_{p s} + i r_{s s}}{\sqrt{2}} \exp (- i k_{r x} δ_{x}^{V} - i k_{r y} δ_{y}^{V}) | - ⟩ | ϕ ⟩ .

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δ_{x}^{H} = \frac{r_{p p}}{k_{0} (r_{p p}^{2} + r_{s p}^{2})} \frac{\partial r_{p s}}{\partial θ_{i}} - \frac{r_{s p}}{k_{0} (r_{p p}^{2} + r_{s p}^{2})} \frac{\partial r_{p p}}{\partial θ_{i}},

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δ_{x}^{V} = \frac{r_{p s}}{k_{0} (r_{s s}^{2} + r_{p s}^{2})} \frac{\partial r_{s s}}{\partial θ_{i}} - \frac{r_{s s}}{k_{0} (r_{s s}^{2} + r_{p s}^{2})} \frac{\partial r_{p s}}{\partial θ_{i}} .

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δ_{y}^{H} = \frac{(r_{p p} + r_{s s}) r_{p p}}{k_{0} (r_{p p}^{2} + r_{s p}^{2})} \cot θ_{i} - \frac{(r_{p s} - r_{s p}) r_{s p}}{k_{0} (r_{p p}^{2} + r_{s p}^{2})} \cot θ_{i},

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δ_{y}^{V} = \frac{(r_{p p} + r_{s s}) r_{s s}}{k_{0} (r_{s s}^{2} + r_{p s}^{2})} \cot θ_{i} + \frac{(r_{p s} - r_{s p}) r_{p s}}{k_{0} (r_{s s}^{2} + r_{p s}^{2})} \cot θ_{i} .

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⟨ x_{r \pm}^{H, V} ⟩ = \frac{⟨ ϕ_{r}^{H, V} | \partial_{k_{r x}} | ϕ_{r}^{H, V} ⟩}{⟨ ϕ_{r}^{H, V} | ϕ_{r}^{H, V} ⟩} .

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⟨ x_{r \pm}^{H} ⟩ = \mp \frac{1}{k_{0}} Re [\frac{r_{p p}}{r_{p p}^{2} + r_{s p}^{2}} \frac{\partial r_{p s}}{\partial θ_{i}} - \frac{r_{s p}}{r_{p p}^{2} + r_{s p}^{2}} \frac{\partial r_{p p}}{\partial θ_{i}}],

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⟨ x_{r \pm}^{V} ⟩ = \mp \frac{1}{k_{0}} Re [\frac{r_{p s}}{r_{s s}^{2} + r_{p s}^{2}} \frac{\partial r_{s s}}{\partial θ_{i}} - \frac{r_{s s}}{r_{s s}^{2} + r_{p s}^{2}} \frac{\partial r_{p s}}{\partial θ_{i}}] .

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⟨ y_{r \pm}^{H, V} ⟩ = \frac{⟨ ϕ_{r}^{H, V} | \partial_{k_{r y}} | ϕ_{r}^{H, V} ⟩}{⟨ ϕ_{r}^{H, V} | ϕ_{r}^{H, V} ⟩} .

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Wenshuai Zhang, Weijie Wu, Shizhen Chen, Jin Zhang, Xiaohui Ling, Weixing Shu, Hailu Luo, Shuangchun Wen. Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals[J]. Photonics Research, 2018, 6(6): 511

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