Loss enhanced spin Hall effect of transmitted light through anisotropic epsilon- and mu-near-zero metamaterial slab

Tingting Tang; Jie Li; Li Luo; Ping Sun; Yanfen Zhang

doi:10.1364/OE.25.002347

1. Introduction

When a linearly polarized beam with a finite width undergoes reflection or transmission at a planar dielectric interface, its left-circularly polarized (LCP) and right-circularly polarized (RCP) components will be split at the interface. The LCP and RCP components of reflected or transmitted beam show a small spatial shift perpendicular to the incidence plane relative to the position predicted by geometrical optics. This phenomenon is called spin Hall effect of light (SHEL) [1–3] which is induced by the spin-orbit momentum coupling to conserve total angular momentum [4–6]. This transverse shift is also called Imbert-Fedorov (IF) shift as it was theoretically predicted by Fedorov [7] and experimentally confirmed by Imbert [8]. SHEL holds a great application potential in precision metrology and quantum information processing [9]. However, the transverse shift of circularly polarized 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 SHEL to facilitate its application [10–12]. Here we notice that the inevitable loss of material is seldom considered in current works as it will bring a great reduction of splitting between RCP and LCP components [13].

Epsilon and mu-near-zero (EMNZ) material is a kind of artificial metamaterial in which both permeability and permittivity are close to zero [14–17]. EMNZ metamaterial can be applied to enhance the directive emission of light [14] and shaping the radiation pattern [15]. It can also be used to realize half mode microwave filters [16] and design perfect coherent antennas [17]. Generally speaking, the loss of EMNZ brings a dramatically reduction of transmitted light. However, Feng proposed a loss enhanced transmission by use of anisotropic epsilon-near-zero (ENZ) metamaterial in 2012 [18]. This prediction is then verified by Sun et al in silver-germanium multilayered structure who also reveal the physical mechanism of this exotic phenomenon [19]. Meanwhile the EMNZ metamaterial with anisotropic permittivity and permeability can also be realized [20–22] which may bring loss enhanced transmission for both p- and s- polarized light.

As the transverse shift of circularly polarized light for vertical polarization input is proportional to the transmission of p-polarized light [23], the loss enhanced transmission inspires us to explore the possibility of loss enhanced SHEL in EMNZ metamaterial. Meanwhile to the best of our knowledge, there is no research about the study of SHEL in EMNZ metamateiral.

In this paper, we first study the loss enhanced SHEL through anisotropic EMNZ metamaterial slab. The physical mechanism of the loss enhanced SHEL is analyzed and the calculation method is deduced in our configuration. Simulation results verify the prediction of loss enlarged transverse shifts of left- and right- circularly polarized beams. The influences of loss and anisotropy on the transverse shift of LCP light are also studied to explore the possible application of this phenomenon.

2. Principles

In our configuration, we assume an incident light is injected into the EMNZ slab from air with an incident angle of θ_i as shown in Fig. 1. The relative permittivity and permeability in region 1-3 are denoted by $ε_{i}$ and $μ_{i}$ (i = 1, 2, 3), respectively. EMNZ metamaterial slab thickness is denoted by $d_{2}$ is anisotropic with a relative permittivity tensor of

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

and a relative permeability tensor of

Fig. 1 Schematic of SHEL through anisotropic EMNZ metamaterial slab

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{\tilde{μ}}_{2} = (\begin{matrix} μ_{∥} \\ μ_{∥} \\ μ_{⊥} \end{matrix})

Based on effective index method, we can obtain the effective refractive indices as $n_{2} = \sqrt{ε_{∥} μ_{∥} - ε_{∥} \sin^{2} θ_{i} / ε_{⊥}}$ , (p-polarized light) and $n_{2} = \sqrt{ε_{∥} μ_{∥} - μ_{∥} \sin^{2} θ_{i} / μ_{⊥}}$ (s-polarized light). The transmission coefficients can be written as [24]

t = \frac{2 n_{2}^{3} k_{0}^{2} \cos θ_{i}}{2 n_{2}^{3} k_{0}^{2} \cos θ_{i} \cos (n_{2} k_{0} d_{2}) + i \sin (n_{2} k_{0} d_{2}) [k_{x}^{2} (1 + n_{2}^{4}) - n_{2}^{2} k_{0}^{2} (1 + n_{2}^{2})]}

in which

k_{x} = k_{0} \sin θ

and

k_{0} = 2 π / λ

. We consider an incident Gaussion beam with an angular spectrum of

{\tilde{E}}_{i} = \frac{w_{0}}{\sqrt{2 π}} \exp [- \frac{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 [23]

[\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, V}

is circular components of the transmitted field. Here H and V represent horizontal and vertical polarization, respectively. Horizontal polarization means polarization in the incident plane which is also called p-polarization. Similarly, vertical polarization means polarization perpendicular to the incident plane which is also called s-polarization.

In our configuration, the incident light is in xoz plane which means the y-component of wave vector (k_y) is equal to zero. In this case the transmitted p-polarized light for s-polarized incident (t_ps) and transmitted s-polarized light for p-polarized (t_sp) both remain zero. Thus the SHEL in EMNZ metamaterial waveguide can be analyzed by the model suitable for isotropic waveguide. On the other hand, it is known to us the left- and right-circularly polarized lights have opposite spin angular momentum when they propagate along the same direction. Once t_ps and t_sp are zero, the angular momentum variations to be compensated by orbit angular momentums are identical for left- or right-circularly polarized beams. It can be predicted that the transverse shifts for left- or right-circularly polarized light are opposite in sign with the same value. Therefore, we can conclude that the separation between left- and right-circularly polarized lights should be symmetric [25] which is similar to the situation in isotropic waveguide.

Therefore the transverse shifts of transmitted light can be defined as [23]

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

in which

δ^{+}

and

δ^{-}

indicate the transverse shift of left- and right-circularly polarized components, respectively. In order to obtain an exact analytical result, we give the spatial transverse shift without neglecting any terms of electric field in the following analysis. By substituting the electrical field distribution into Eq. (6), the spatial transverse shifts through the EMNZ slab 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} + {(\frac{\partial t_{p}}{\partial θ_{i}})}^{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} + {(\frac{\partial t_{s}}{\partial θ_{i}})}^{2}}

From Eq. (6), we can find the transverse shift of SHEL is determined by the electrical filed distribution of transmitted light through EMNZ metamaterial slab which is proportional to $| t_{p} | / | t_{s} |$ or $| t_{s} | / | t_{p} |$ . It can be deduced that a large (small) transmission of p-polarized light and a small (large) transmission of s-polarized light will result in a large transverse shift of circularly-polarized light for V (H) polarization input. In the EMNZ metamaterial waveguide, the transmittance of p-polarized light is dramatically enhanced with the increase of $Im (ε_{⊥})$ [19]. As the splitting between left- and right circularly polarized light is closely related to the transmittance of p- and s-polarized light beams, it is easy to find the material loss can be used to modulate the SHEL in EMNZ metamaterial waveguide. By choosing proper metamaterial loss, the transmittances of p- and s-polarized light beams can satisfy the SHEL enhancement conditions in Eqs. (7) and (8). In this case, the metamaterial loss will affect the spin-orbit coupling of transmitted light. It means the angular momentum variations to be compensated by orbit angular momentums are modulated by metamaterial loss. Thus the spin-orbit coupling can be controlled by the metamaterial loss. This is the basic principle of loss enhanced SHEL in this paper.

To illustrate this principle, we give the transverse shift contours for H and V polarization inputs with different $Im (ε_{⊥})$ in Fig. 2(a) and (b), respectively. We choose λ = 1550 nm, d₂ = 2λ, $ε_{∥} = 1$ , $Re (ε_{⊥}) = 0.01$ , $μ_{∥} = 1$ and $μ_{⊥} = 0.01 + 0.01 i$ . For simplicity we assume the transverse shift to be multiples of wavelength which will not be repeated in the following discussions. The transverse shift of LCP for H polarization input is enlarged when the loss of $ε_{⊥}$ is smaller than 0.01. Positive and negative peaks of transverse shift appear at $θ > 6^{\circ}$ which achieve ± 15λ. When $Im (ε_{⊥})$ is larger than 0.01, there is no obviously enlarged transverse shift in the angular spectrum. For V polarization input, the results are totally different. When $Im (ε_{⊥})$ is smaller than 0.2, enhanced SHEL does not appear. While for $Im (ε_{⊥}) > 0.1$ , there is a flat-top transverse shift peak when the incident angle is in a range from 6 to10 degrees. We also find when $0.2 < Im (ε_{⊥}) < 0.4$ , the transverse shift peak has a shift to a smaller incident angle with the increase of loss. Then with the further increase of $Im (ε_{⊥})$ , transverse shift peaks remain unchanged. As the increase of loss brings no obvious change of the transmittances of s- and p-polarized lights, $| t_{p} | / | t_{s} |$ also remains unchanged. Thus the transverse shift of LCP light is invariant with EMNZ metamaterial loss.

Fig. 2 Transverse shift contour (integer multiples of wavelength) of LCP light for different $Im (ε_{⊥})$ with H (a) and V (b) inputs, respectively. Here we choose d₂ = 2λ, $ε_{∥} = 1$ , $Re (ε_{⊥}) = 0.01$ , $μ_{| |} = 1$ , $μ_{⊥} = 0.01 + 0.01 i$ .

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3. Results and discussion

In this section, we explore the transverse shift distribution on angular spectrum for different $Re (ε_{⊥})$ , $Re (μ_{⊥})$ , $Im (μ_{⊥})$ and d₂. We first choose λ = 1550 nm, d₂ = 2λ, $μ_{∥} = 1$ and $μ_{⊥} = 0.01 + 0.01 i$ , the transverse shift contours of LCP light for different with V input are shown in Fig. 3. When $Im (ε_{⊥}) = 0.001$ , there are five transverse shift peaks as shown in Fig. 3(a). For $Re (ε_{⊥}) = 0.02$ , a negative transverse shift appears at around which achieves −14.08λ. With the increase of $Re (ε_{⊥})$ , this transverse shift peak is dramatically weakened. Meanwhile a positive peak of 15.76λ and a negative peak of −14.85λ appear at $θ = {12.24}^{\circ}$ and $θ = {17.49}^{\circ}$ , respectively. For $Re (ε_{⊥}) = 0.01$ these transverse shift peaks have obvious shifts to larger incident angles as shown in Fig. 3(b). The negative peak at $θ = {6.33}^{\circ}$ is slightly enlarged to 15.55λ. However, the peaks of transverse shift at $θ = {9.96}^{\circ}$ and $θ = {16.16}^{\circ}$ are reduced to 9.02λ and −5.90λ, respectively. In Fig. 3(c), the $Im (ε_{⊥})$ is increased to 0.1, we can find all the positive and negative transverse shift peaks are merged into one positive peak at around $θ = 6^{\circ}$ . With the increase of $Re (ε_{⊥})$ the peak value is slightly reduced and the corresponding incident angle has a small shift to large incident angle. As the transverse shift contour looks like a band-pass filter, a flat-top transverse shift can be observed on the angular spectrum. It can also be considered as a strong tolerance to variation of $Im (ε_{⊥})$ which has not been observed in other waveguides to our knowledge up to now. This property can be expected to realize new optoelectronic devices with anti-interference performance.

Fig. 3 Transverse shift contour (integer multiples of wavelength) of LCP light for different $Re (ε_{⊥})$ with V input including $Im (ε_{⊥}) = 0.001$ (a), $Im (ε_{⊥}) = 0.01$ (b) and $Im (ε_{⊥}) = 0.1$ (c). Here we choose d₂ = 2λ, $μ_{∥} = 1$ , $μ_{⊥} = 0.01 + 0.01 i$ .

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Then we take into account the influences of $μ_{⊥}$ on the SHEL in EMNZ slab. We assume the real and imaginary parts are identical which means $μ_{⊥} = α + α i$ . The transverse shift contours of LCP light for different α with V input are shown in Fig. 4. When $Im (ε_{⊥}) = 0.001$ , a weak flat-top transverse shift peak of less than 5λ can be observed if as shown in Fig. 4(a). With the decrease of α, six positive and negative peaks can be found. When α = 0.005, the maximum positive transverse shift is 15.38 λ ( $θ = {9.41}^{\circ}$ ) and the negative one is −15.40 λ ( $θ = {13.2}^{\circ}$ ). When $Im (ε_{⊥}) = 0.01$ , the transverse shift peaks appear at α = 0.005 are merged together. For larger α = 0.007, the negative peak achieves −15.85λ at $θ = {7.8}^{\circ}$ as shown in Fig. 4(b). When $Im (ε_{⊥})$ is increased to 0.1, there is one transverse shift peak as shown in Fig. 4(c). With the increase of α, the incident angle corresponding to the transverse shift peak also increases which is determined by the phase-matching condition. Meanwhile the transverse shift band is also broadened. Here the maximum transverse shift of 15.92λ can be found at α = 0.018 and $θ = {9.41}^{\circ}$ .

Fig. 4 Transverse shift contour (integer multiples of wavelength) of LCP light for different α with V input including $Im (ε_{⊥}) = 0.001$ (a), $Im (ε_{⊥}) = 0.01$ (b) and $Im (ε_{⊥}) = 0.1$ (c). Here we choose d₂ = 2λ, $ε_{∥} = 1$ , $Re (ε_{⊥}) = 0.01$ and $μ_{∥} = 1$ .

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At last, we explore the transverse shift of LCP light in EMNZ slab with different thickness as shown in Fig. 5. We also find there are several transverse shift peaks on the angular spectrum in Fig. 5(a) when $Im (ε_{⊥}) = 0.001$ .When the incident angle is 8 degrees, five periodic positive and negative peaks appear at different EMNZ slab thickness. When $Im (ε_{⊥}) = 0.01$ , three transverse shift peaks are greatly broadened in Fig. 5(b). For d₂ = 3λ, three peaks of −11.43λ, 15.87λ and −14.53λ appear at ${6.15}^{\circ}$ , ${9.75}^{\circ}$ and ${17.62}^{\circ}$ , respectively, respectively. When $Im (ε_{⊥}) = 0.1$ , these transverse shift peaks merge into one as shown in Fig. 5(c). The flat-top transverse shift peak has a small shift to a smaller incident angle and the width of pass band remains almost unchanged with the increase of EMNZ slab thickness. Thus we can also conclude that in the proposed configuration, the enhanced SHEL effect in EMNZ slab is insusceptible to the thickness change. As is known to us, EMNZ metamaterial is always realized by multilayers of different films, it is very difficult to make the thickness smaller than 1 μm. Generally a thicker EMNZ slab will induce more absorption and then dramatically weaken SHEL effect [13]. In our structure, however, the loss of EMNZ slab can enhance the SHEL effect due to the anisotropic configuration of metamaterial permittivity and permeability. This brings us a much easier fabrication method to obtain large splitting of left- and right- circularly polarized light in EMNZ slab.

Fig. 5 Transverse shift contour (integer multiples of wavelength) of LCP light for different $Im (μ_{⊥})$ with V input including $Im (ε_{⊥}) = 0.001$ (a), $Im (ε_{⊥}) = 0.01$ (b) and $Im (ε_{⊥}) = 0.1$ (c). Here we choose $ε_{∥} = 1$ , $Re (ε_{⊥}) = 0.015$ , $μ_{∥} = 1$ and $μ_{⊥} = 0.01 + 0.01 i$ .

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4. Conclusion

In this paper, we propose the loss enhanced SHEL through anisotropic EMNZ metamaterial slab. In contrast to the normal situation that the loss will greatly weaken the transmitted light, the loss of EMNZ metamaterial can effectively increase the transmission of p-polarized light when the incident angle is much larger than the critical angle in our configuration. As is proportional to the transmitted light of p-polarized light, the splitting of RCP and LCP can be greatly enlarged when light of V polarization is transmitted through the anisotropic EMNZ slab. Thus the loss of EMNZ metamaterial does not weaken but enhance SHEL by anisotropic configuration of epsilon and mu tensors. When the imaginary part of permeability’s vertical component is equal to 0.1, a flat-top transverse shift peak can be observed which remains unchanged for different $ε_{⊥}$ and d₂. In this case the maximum transverse shift of LCP light for V input can be increased to 24.676 micrometers without any amplification method. As it makes quantum devices based on SHEL available, our work paves the way towards on-chip and inter-chip optical circuitry.

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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Loss enhanced spin Hall effect of transmitted light through anisotropic epsilon- and mu-near-zero metamaterial slab

Abstract

1. Introduction

2. Principles

3. Results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (8)

Optics Express