Spin Hall effect of transmitted light in a three-layer waveguide with lossy epsilon-near-zero metamaterial

Tingting Tang; Jie Li; Yanfen Zhang; Chaoyang Li; Li Luo

doi:10.1364/OE.24.028113

1. Introduction

When a linearly polarized beam is incident to a planar dielectric interface the left- and right-circular components will be split which show a small transverse shift perpendicular to the incident plane. This phenomenon is known as spin Hall effect (SHE) of light which is induced by the spin-orbit momentum coupling based on total angular momentum conservation law [1–4]. The transverse shift is also called Imbert-Fedorov (IF) as it was theoretically predicted by Fedorov [5] and experimentally confirmed by Imbert [6]. It holds great potential applications in precision metrology and quantum information processing. Generally the transverse shift of SHE of transmitted light is on the subwavelength scale and is difficult to be directly measured with conventional experimental methods. Therefore in recent years, researchers take measures to enhance and modulate the IF shift to facilitate its application. Large splitting of right- and left-circularly polarized waves of reflected light at metasurfaces [7] or chiral metamaterial [8] can be realized. In surface plasmon resonance (SPR) waveguide a sharp dip in the reflected light intensity may induce an enlarged transverse shift of circularly polarized light [9, 10]. The SHE of reflected light can also be enhanced by long range surface plasmon resonance (LRSP) and electro-optically modulated in a prism-waveguide coupling system [11]. For transmitted light, SHE of transmitted light in photonic tunneling is proposed by Luo [12] and is then observed via weak measurements in a three-layer barrier structure [13] which brings a possibility of spin-based nano-photonic applications [14, 15]. These achievements may provide a route for exploiting the spin and orbit angular momentum of light for information processing and communication.

Epsilon-near-zero (ENZ) metamaterial is a kind of artificial medium in which permittivity is close to zero. ENZ metamaterial can be applied to enhance the directive emission of light [16] and shaping the radiation pattern [17]. It can also be used to realize energy collimation [18, 19]. As metamaterial is always lossy, transmitted light in ENZ metamaterial is inevitablly reduced by large absorption. However, for anisotropic ENZ metamaterial, the situation is totally different. In 2012, Feng proposed a loss enhanced transmission by use of anisotropic epsilon-near-zero (ENZ) metamaterial which is induced by corresponding decreased propagation loss for oblique incidence due to the increased material loss [20]. This prediction is then verified by Sun et al in silver-germanium multilayered structure [21]. It is known that the transverse shift of circularly polarized light for vertical polarization input is proportional to the transmission of p-polarized light [22]. Therefore the loss enhanced transmission provides us a new method to enhance SHE of transmitted light in ENZ metamaterial waveguide.

In this paper, we study the SHE of transmitted light in a three-layer waveguide with a lossy ENZ metamaterial. Theoretical analysis and physical interpretations are given based on simulation results. The influences of ENZ permittivity components and gold layer thicknesses on the SHE of transmitted light through air/ENZ/Au/ENZ/air waveguide are also analyzed.

2. Theoretical analysis

We assume an incident light is injected into the ENZ/Au/ENZ waveguide from air with an incident angle of θ. For convenience we assume every layer of the waveguide as well as air layers is numbered as shown in Fig. 1. The relative permittivity and permeability of materials in region 1-5 are denoted by $ε_{i}$ and $μ_{i}$ (i = 1, 2, 3, 4, 5), respectively. The thicknesses of ENZ metamaterial, gold layer and ENZ metamaterial are denoted by $d_{2}$ , $d_{3}$ and $d_{4}$ , respectively. ENZ metamaterial has an anisotropic permittivity tensor of

Fig. 1 Schematic for SHE of transmitted light in an ENZ-gold-ENZ waveguide placed in air.

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{\tilde{ε}}_{2} = (\begin{matrix} ε_{| |} \\ ε_{| |} \\ ε_{⊥} \end{matrix})

The reflective coefficients of each interface can be written as [19]

r_{2} = \frac{r_{12} + r_{3} \exp (2 i k_{2 z} d_{2})}{1 + r_{12} r_{3} \exp (2 i k_{2 z} d_{2})}

r_{3} = \frac{r_{23} + r_{4} \exp (2 i k_{3 z} d_{3})}{1 + r_{23} r_{4} \exp (2 i k_{3 z} d_{3})}

r_{4} = \frac{r_{34} + r_{45} \exp (2 i k_{4 z} d_{4})}{1 + r_{34} r_{45} \exp (2 i k_{4 z} d_{4})}

The transmission coefficient of light through the ENZ/Au/ENZ waveguide can be written as

t = t_{12} t_{23} t_{34} t_{45} \exp (i k_{2 z} d_{2}) \exp (i k_{3 z} d_{3}) \exp (i k_{4 z} d_{4})

For s-polarized light we have

r_{i j} = \frac{k_{i z} - k_{j z}}{k_{i z} + k_{j z}}

,

t_{i j} = \frac{2 k_{i z}}{k_{i z} + k_{j z}}

,

k_{i z} = \sqrt{k_{0}^{2} ε_{i} μ_{i} - k_{x}^{2}} (i = 1, 3, 5)

and

k_{j z} = \sqrt{ε_{| |} μ_{j} - k_{x}^{2}} (j = 2, 4)

.

For p-polarized light we have $r_{i j} = \frac{k_{i z} / ε_{i} - k_{j z} / ε_{j}}{k_{i z} / ε_{i} + k_{j z} / ε_{j}}$ and $t_{i j} = \frac{2 k_{i z} / ε_{i}}{k_{i z} / ε_{i} + k_{j z} / ε_{j}}$ , $k_{i z} = \sqrt{k_{0}^{2} ε_{i} μ_{i} - k_{x}^{2}} (i = 1, 3, 5)$ and $k_{j z} = \sqrt{ε_{| |} μ_{| |} - ε_{| |} k_{x}^{2} / ε_{⊥}} (j = 2, 4)$ .

We consider an incident Gaussion beam with an angular spectrum of ${\tilde{E}}_{i} = w_{0} / \sqrt{2 π} \exp [- w_{0}^{2} (k_{x}^{2} + k_{y}^{2}) / 4]$ in which w₀ is the beam waist. In our configuration as shown in Fig. 1, the incident plane is x-z and the transverse shift is along y-axis. Thus the angular spectra between transmitted and incident light beams can be written as [20]

[\begin{matrix} {\tilde{E}}_{t}^{H} \\ {\tilde{E}}_{t}^{V} \end{matrix}] = [\begin{matrix} t_{p} & k_{y} (t_{p} - t_{s}) \cot (θ_{i}) \\ k_{y} (t_{p} - t_{s}) \cot (θ_{i}) & t_{s} \end{matrix}] [\begin{matrix} {\tilde{E}}_{i}^{H} \\ {\tilde{E}}_{i}^{V} \end{matrix}]

in which

{\tilde{E}}_{t}^{H}

and

{\tilde{E}}_{t}^{V}

denote the horizontal and vertical components of angular spectra of transmitted beams components. Here H and V represent horizontal and vertical polarizations, respectively. Horizontal polarization means polarization in the plane of incidence and is the synonym of p-polarization. Similarly, vertical polarization means polarization perpendicular to the plane of incidence and is the synonym of s-polarization.

The transverse shifts of transmitted light can be defined as [21]

δ_{H, V}^{\pm} = \frac{\iint y {| E_{H, V}^{\pm} |}^{2} d x d y}{\iint {| E_{H, V}^{\pm} |}^{2} d x d y}

in which

δ^{+}

and

δ^{-}

indicate the transverse shifts of left- and right-circularly polarized components, respectively. By neglecting the terms with derivatives, the spatial transverse shifts through the air/ENZ/Au/ENZ/air waveguide can be obtained as

δ_{H}^{\pm} = \pm \frac{k_{0} w_{0}^{2} \cot θ_{i} ({| t_{p} |}^{2} \frac{\cos θ_{t}}{\cos θ_{i}} - Re (t_{p} \cdot t_{s}^{*}))}{k_{0}^{2} w_{0}^{2} {| t_{p} |}^{2} + \cot^{2} θ_{i} {| t_{p} \frac{\cos θ_{t}}{\cos θ_{i}} - t_{s} |}^{2}}

δ_{V}^{\pm} = \pm \frac{k_{0} w_{0}^{2} \cot θ_{i} ({| t_{s} |}^{2} \frac{\cos θ_{t}}{\cos θ_{i}} - Re (t_{s} \cdot t_{p}^{*}))}{k_{0}^{2} w_{0}^{2} {| t_{s} |}^{2} + \cot^{2} θ_{i} {| t_{s} \frac{\cos θ_{t}}{\cos θ_{i}} - t_{p} |}^{2}}

When a linear polarized light is transmitted through the three-layer waveguide, its left- and right-circularly polarized components have transverse shifts in y-axis. It is known to us the left- and right-circularly polarized light have opposite spin angular momentum when they propagate along the same direction. Meanwhile the angular momentum variations to be compensated by orbit angular momentums are identical for them. It can be predicted that the transverse shifts for left- or right-circularly polarized light are opposite in sign and with the same value. That is to say the splitting between left- and right-circularly polarized light should be symmetric.

3. Simulation results and analysis

In this section, we explore the SHE of transmitted light in an air/ENZ/Au/ENZ/air waveguide. We choose λ = 1550 nm, $Re (ε_{⊥}) = 0.01$ , $ε_{| |} = 1$ , $ε_{3} = - 96.958 + 11.503 i$ , d₂ = d₄ = 1 μm and d₃ = 20 nm. For convenience we define a figure of merit (FOM) which is equal to $Re (ε_{⊥}) / Im (ε_{⊥})$ . The transmittances of p- and s- polarized light are shown in Figs. 2 (a) and 2(b), respectively. It is easy to see with the decrease of FOM, the transmittance is dramatically broadened in angular spectrum. It is to say with the increase of imaginary part of $ε_{⊥}$ the incident light with a larger range of incident angle can transmit through the ENZ/Au/ENZ waveguide. In an ordinary case, the loss of material will greatly reduce the SHE of transmitted light [23]. However, instead of preventing the transmission of light, the loss of ENZ metamaterial will help to increase the transmission in this case. This phenomenon is induced by corresponding decreased propagation loss for oblique incidence due to the increased material loss [20]. This result is consistent with that in [21]. On the other hand, the transmittance of s-polarized light is not affected by FOM which is associated with $μ_{2}$ rather than $Im (ε_{⊥})$ .

Fig. 2 Transmittances of p-polarized light (a) and s-polarized light (b) for different FOM.

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Then we calculate the transverse shift of left-circularly polarized light with vertical polarization input for different $Re (ε_{⊥})$ and $Im (ε_{⊥})$ as shown in Fig. 3. Here we assume $Im (ε_{⊥})$ changes in a range from 0.001 to 0.02. We also choose λ = 1550 nm, $ε_{| |} = 1$ , d₂ = d₄ = 1 μm and d₃ = 20 nm. When $Re (ε_{⊥}) = 0.001$ , a large negative transverse shift peak of −30λ appears close to $Im (ε_{⊥}) = 0.001$ as shown in Fig. 3(a). With the increase of $Im (ε_{⊥})$ this negative transverse shift peak disappears and a positive peak appears. We can find the positive peak is a flat-top band which is broadened as the increase of $Im (ε_{⊥})$ . Meanwhile the center of the transverse shift peak has a smaller shift to a larger incident angle.

Fig. 3 Transverse shift contours (integer multiples of wavelength) of left-circularly polarized light with vertical polarization input for different $Re (ε_{⊥})$ and $Im (ε_{⊥})$ .

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When $Re (ε_{⊥}) = 0.005$ , a negative transverse shift peak can be found when $Im (ε_{⊥})$ is between 0.001 and 0.002 as shown in Fig. 3(b). The flat-top band of positive transverse shift is also broadened. We also notice that when $Im (ε_{⊥}) = 0.002$ a positive peak of 15λ and a negative peak of −10λ can be observed at the same time. With the increase of $Re (ε_{⊥})$ , more and more double-peak transverse shifts appear in Figs. 3(c) and 3(d). Meanwhile the peak values of transverse shift are reduced. As the transverse shift is associated with the electrical filed distribution as shown in Eq. (7), this can be explained as that the transverse shift is proportional to the distribution of electrical filed. It is known to us the total electrical field remains unchanged, thus more transverse shift peaks will certainly reduce the peak value of electrical field. Thus the transverse shift peak value is smaller for $Re (ε_{⊥}) = 0.02$ than that for $Re (ε_{⊥}) = 0.01$ as two obvious peaks appear in Fig. 3(d).

In order to make our results convincing we use anisotropic ENZ metamaterial consisting of alternating thin layers of silver and germanium stacking [21]. According to the effective medium theory, the permittivity components can be written as

ε_{| |} = \frac{1}{f / ε_{A g} + (1 - f) / ε_{G e}}

ε_{⊥} = f ε_{A g} + (1 - f) ε_{G e}

in which

f = 0.1432

,

ε_{A g} = 5 - ω_{p}^{2} / (ω^{2} + i α γ ω)

,

γ = 5.07 \times 10^{13} r a d / s

,

α = 7

ω_{p} = 1.38 \times 10^{16} rad / s

and

ε_{G e} = 19.01 + 0.087 i

. When the incident wavelength is 1.55 μm (

ω = 1.2161 \times 10^{15} rad / s

), we have

ε_{| |} = 22.7664 + 0.289 i

and

ε_{⊥} = 0.0109 + 5.033 i

. We first give the transmittances of p-and s-polarized light for different ENZ metamaterial thickness as shown in Figs. 4(a) and 4(b), respectively. Here we choose d₃ = 20 nm and d₄ = d₂. With the increase of d₂, the transmission of light has a fluctuation when the incident angle is smaller than 20 degrees. The transmission depends on d₂ when angle of incidence θ < 20°. For higher angles of incidence, the transmission curves for different d₂ merge together. We also find the transmission of light achieves 0.96 when incident angle is equal to 77.5 degrees. For s-polarized light, its transmission decreases as the increase of incident angle. Meanwhile we find when d₂ >5 μm the transmission curves show an oscillatory property as in Fig. 4(b).

Fig. 4 Transmittances of p-polarized light (a) and s-polarized light (b) for different ENZ metamaterial thickness

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The transverse shift contours of left-circularly polarized light with horizontal and vertical polarization inputs for different ENZ metamaterial thickness are shown in Figs. 5 (a) and 5(b), respectively. Here we also choose the same parameters as that in Fig. 4. For horizontal polarization input, a large positive transverse shift peak appears when the incident angle is close to zero. It almost remains unchanged for different ENZ metamaterial thickness. It can be observed that when incident angle is close to zero, there is a narrow transverse shift peak in Fig. 5(a). This phenomenon occurs when the incident light is partially transmitted from the ENZ/Au/ENZ waveguide. As the real part of permittivity for ENZ metamaterial is close to zero, the transverse shift peak in this case also appears at a rather small incident angle. When d₂ is larger than 4 μm another negative transverse shift peak can be found at around $θ = 70^{\circ}$ which is enlarged and has a shift to smaller incident angle with the increase of ENZ metamaterial thickness. In addition, a positive transverse shift peak appears at around 85 degrees when d₂ = 5 μm. Generally speaking, a transverse shift peak can be observed when the incident angle is less than 20 degrees because of a small critical angle in ENZ metamaterial waveguide. The anisotropy of ENZ metamaterial waveguide makes it possible to find a transverse shift peak when $θ > 20^{\circ}$ . Meanwhile the transverse shift peak appears at around 85 degrees is induced by SPR excited at the interface between ENZ metamaterial and gold layers.

Fig. 5 Transverse shift contours (integer multiples of wavelength) of left-circularly polarized light with horizontal polarization (a) and vertical polarization (b) inputs for different ENZ metamaterial thickness.

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For vertical polarization input, the situation is much simpler as shown in Fig. 5(b). There is only one obvious flat-top transverse shift peak for different ENZ metamaterial thickness. For d₂ = 1 μm a large transverse shift can be observed when the incident angle is between 49 degrees and 69 degrees. With the increase of d₂, this transverse shift band is gradually narrowed and has a shift to smaller incident angle. This flat-top transverse shift peak is induced by the transmittance which is almost unchanged with the ENZ metamaterial thickness. It can be concluded that the transverse shift is still determined by the transmittance of light. Thus by controlling transmission of incident light, the SHE of transmitted light can be enhanced and modulated flexibly.

At last, we study the influence of gold layer thickness on the transmittance and transverse shift when it equals to 10 nm, 15 nm, 20 nm and 25 nm. Transmittances of p-polarized light and s-polarized light for different gold layer thickness are shown in Figs. 6(a) and 6(b), respectively. The transmittance for smaller incident angle less than 20 degrees decreases as the increase of gold layer thickness. Then the four transmission curves merge together and achieve a maximum value of 0.98 at around 76 degrees. For s-polarized light the transmission is reduced by a thicker gold layer which also decreases with the incident angle. In Fig. 7, we give the transverse shift of left-circularly polarized light with horizontal polarization (a) and vertical polarization (b) inputs for different gold layer thickness. It is easy to see the transverse shift distribution will not be affected by the gold layer thickness. It can be explained as that the loss enhanced transmission constructs a photonic tunneling structure which is invariant with the increase of the golden interlayer thickness. This provides us a good tolerance to the gold thickness in experiment. Meanwhile we can find the transverse shift peak with horizontal polarization input appears at an incident angle near zero. For vertical polarization input, the large transverse shift peak is induced by the transmission peak for p-polarized light. Thus the transverse shift peak appears at about 60 degrees. In addition, we can find the maximum transverse shifts of left-polarized light for horizontal and vertical polarization inputs achieve 32λ (49.6 microns) and 22λ (34.1 microns), respectively.

Fig. 6 Transmittances of p-polarized light (a) and s-polarized light (b) for different gold layer thickness.

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Fig. 7 Transverse shift of left-circularly polarized light with horizontal polarization (a) and vertical polarization (b) inputs for different gold layer thickness.

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4. Conclusion

In this paper, we study the SHE of transmitted light in an ENZ/Au/ENZ waveguide. Based on simulation results, the influence of ENZ permittivity components and thickness as well as gold layer thickness on transverse shift of left-circularly polarized light is analyzed. We find the loss of anisotropic ENZ metamaterial brings a different distribution of transverse shift peaks. To make our results convincing, we make use of anisotropic ENZ metamaterial consisting of alternating thin layers of silver and germanium stacking to construct the ENZ/Au/ENZ waveguide. The transverse shifts for horizontal and vertical polarization inputs with different ENZ metamaterial thickness and gold layer thickness are obtained. In this case, the maximum transverse shifts of left-polarized light for horizontal and vertical polarization inputs achieve 49.6 and 34.1 microns, respectively. Meanwhile the enhanced SHE of transmitted light is invariant with the variation of gold layer which shows a great tolerance to fabrication error.

Funding

National Natural Science Foundation of China (NSFC) (61505016); Project of Sichuan Provincial Department of Education (15ZA0183); Scientific research fund of Chengdu University of Information Technology (J201417).

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Spin Hall effect of transmitted light in a three-layer waveguide with lossy epsilon-near-zero metamaterial

Abstract

1. Introduction

2. Theoretical analysis

3. Simulation results and analysis

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (11)

Optics Express