Spin Hall effect of reflected light in dielectric magneto-optical thin film with a double-negative metamaterial substrate

Jie Li; Tingting Tang; Li Luo; Nengxi Li; Pengyu Zhang

doi:10.1364/OE.25.019117

1. Introduction

Classical geometrical optics predicts that both transmitted and reflected light are in the plane of incidence when a linearly polarized light beam is incident to a planar dielectric interface. But in fact, in order to satisfy the conservation of angular momentum of light, the left- and the right-handed circularly polarized (LHCP and RHCP) components in the reflected and transmitted light beam will experience a transverse shift (the centroid shifts of the left- and the right-handed circularly polarized light perpendicular to the incident plane) in opposite directions. This phenomenon is called as spin Hall effect of light (SHEL) [1]. In recent years, people have studied the SHEL theoretically or experimentally in various structures. Many methods have been proposed to realize the enhancement and control of spin-dependent splitting of SHEL, such as using metamaterials [2–5], metal surface plasmon structure [6–8] and semiconductor microcavity [9,10]. On the other hand, research on the SHEL has led to a range of applications in precision metrology [11–13].

A metamaterial means an engineered electromagnetic structure with some unique electromagnetic properties over certain frequency bands. They can display negative effective permittivity and/or negative effective permeability over certain frequency bands when they are designed properly rather than a specific chemical compositions. A material with either a negative permittivity or a negative permeability is known as a single-negative (SNG) medium [14]. Metamaterials with both negative permittivity and permeability are called double-negative (DNG) media [15].

Magneto-optical (MO) effect is one of the classical physical phenomena that breaks the symmetry of time inversion and has been widely used in industrial fields. The most common applications are magneto-optical recording [16,17] and magneto-optical isolation [18–20]. When linearly polarized light reflects from a magnetic material, angular momentum is transferred to reflected waves and induces a rotation of polarization plane, this phenomenon is called magneto-optical Kerr effect (MOKE). According to the direction of external magnetic field or magnetization, there are three types of magneto-optical Kerr effect, inlcuding polar magneto-optical Kerr effect (PMOKE), transverse magneto-optical Kerr effect (TMOKE) and longitudinal magneto-optical Kerr effect (LMOKE). From a macroscopic point of view, PMOKE and LMOKE behave as a nonreciprocal modulation in the polarization of reflected light, which is called Kerr rotation. TMOKE appears as a nonreciprocal phase shift (NRPS). Xiaodong Qiu et al. have shown that additional spin-orbit coupling can be formed due to Kerr rotation, which modulates the SHEL shift in turn [11]. As MOKE is very weak in ordinary magnetic materials, many ideas have been proposed to enhance it, including structures based on surface plasmon resonance (SPR) [21–27], extraordinary optical transmission (EOT) [28,29], and metamaterials [30–32]. In general, SPR and EOT configurations are usually accompanied by high optical losses due to high optical damping of metal nanostructures. However, optical losses in metamaterials can be reduced by a reasonable structure since it is flexible to design them [33]. Meanwhile, magneto-optical effect can be greatly enhanced by metamaterials [30,31]. The enhancement of magneto-optical Kerr effect may lead to an enhanced modulation of SHEL by magnetic field.

In this paper, we study SHEL in a dielectric magneto-optical thin film of Ce₁Y₂Fe₅O₁₂ (Ce:YIG) with a double-negative metamaterial substrate. Both theoretical analysis and physical interpretations are given based on simulation results. As a comparison, a thin film of Ce:YIG with a single-negative (SNG) substrate is also analyzed. To make our results convincing we use a realizable DNG metamaterial with silver nanostructures as substrate to verify our conclusion at last.

2. Theoretical analysis

For convenience we assume every layer of the structure as well as air cladding is numbered as shown in Fig. 1. We assume that Gaussian beam is incident from air to the Ce:YIG layer with an angle of $θ_{i}$ in xoz plane. The substrate is composed with double-negative metamaterials and has a semi-infinitely thickness. Due to SHEL, the LHCP and RHCP components will be split in y-direction when the incident light is reflected on the upper surface of the magnetic layer. The relative permittivity and permeability of the first and third layer are expressed as $ε_{i}$ and $μ_{i} (i = 1, 3)$ . The relative permeability (permittivity) of MO layer is denoted by $μ_{2}$ ( ${\overset{\leftrightarrow}{ε}}_{2}$ ), and the thickness is denoted by d₂. The three red arrows in the figure describe three directions of the saturation magnetization vector M of the magnetic layer, which represent PMOKE, TMOKE and LMOKE configurations, respectively.

Fig. 1 Schematic for SHE of reflected light in a magnetic thin film with DNG substrate. The red arrows is the magnetization direction of three types of MOKE configurations.

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Without losing generality, we assume that the applied magnetic field is along x-axis (LMOKE) and satisfies the saturated magnetization condition of Ce:YIG. Then the permittivity tensor of Ce:YIG can be written as

{\overset{\leftrightarrow}{ε}}_{2} = [\begin{matrix} ε_{20} & 0 & 0 \\ 0 & ε_{20} & i ε_{21} \\ 0 & - i ε_{21} & ε_{20} \end{matrix}]

in which

μ_{2} = 1

. A plane wave propagating in the nth (n = 1,2,3) layer

E^{(n)} = E_{0}^{(n)} \exp [i (ω t - γ^{(n)} \cdot r)]

satisfies the equation of

γ^{(n) 2} E_{0}^{(n)} - γ^{(n)} (γ^{(n)} \cdot E_{0}^{(n)}) = \frac{ω^{2}}{c^{2}} {\overset{\leftrightarrow}{ε}}^{(n)} μ^{(n)} E_{0}^{(n)}

where

γ^{(n)} = ω (\hat{x} N_{x} + \hat{z} N_{z}) / c

is wave vector in the nth layer,

E_{0}^{(n)}

is amplitude of electric field. According to the boundary conditions of electromagnetic field, the parallel components (x component) of the wave vector in each layer are equal. By solving Eq. (2), we can obtain the vertical components of each wave vector,

ω N_{z j}^{(n)} / c (j = 1, 2, 3, 4)

. In an isotropic layer,

N_{z j}^{(n)} = N_{z 0}^{(n)} (j = 1, 2, 3, 4)

, in which

N_{z 0}^{(n)} = N_{}^{(n) 2} - N_{x}^{2}

,

N_{}^{(n) 2} = μ^{(n)} ε_{0}^{(n)}

. In the magnetic layer, the elements of the dynamic matrix are [34–36].Then the dynamic matrix is

D^{(2)} = [\begin{matrix} D_{11} & - D_{11} & D_{13} & - D_{13} \\ D_{21} & D_{21} & D_{23} & D_{23} \\ D_{31} & D_{31} & D_{33} & D_{33} \\ D_{41} & - D_{41} & D_{43} & - D_{43} \end{matrix}]

where

D_{11} = i ε_{21} N_{x} N_{z 1}, D_{13} = i ε_{21} N_{x} N_{z 2}, D_{21} = D_{11} N_{z 1}, D_{23} = D_{13} N_{z 2},

D_{31} = (ε_{20} - N_{x}^{2}) (ε_{20} - N_{x}^{2} - N_{z 1}^{2}) + ε_{21}^{2}, D_{33} = (ε_{20} - N_{x}^{2}) (ε_{20} - N_{x}^{2} - N_{z 2}^{2}) + ε_{21}^{2},

D_{41} = N_{z 3} [ε_{20} (ε_{20} - N_{x}^{2} - N_{z 1}^{2}) + ε_{21}^{2}] a n d D_{43} = N_{z 4} [ε_{20} (ε_{20} - N_{x}^{2} - N_{z 2}^{2}) + ε_{21}^{2}] .

The propagation matrix of isotropic layers and magnetic layer is

P^{(n)} = [\begin{matrix} e^{i (ω / c) N_{z 1}^{(n)} d^{(n)}} & 0 & 0 & 0 \\ 0 & e^{i (ω / c) N_{z 2}^{(n)} d^{(n)}} & 0 & 0 \\ 0 & 0 & e^{i (ω / c) N_{z 3}^{(n)} d^{(n)}} & 0 \\ 0 & 0 & 0 & e^{i (ω / c) N_{z 4}^{(n)} d^{(n)}} \end{matrix}]

in which d⁽ⁿ⁾ is the thickness of nth layer. Combining the D matrix and P matrix of each layer, we can get a Q matrix which connects the electric field amplitudes of cover and substrate

Q = D^{{(1)}^{- 1}} D^{(2)} P^{(2)} D^{(2)}^{- 1} D^{(3)}

Then the reflection coefficients of a multilayer structure can be calculated as

r_{s s} = \frac{Q_{21} Q_{33} - Q_{23} Q_{31}}{Q_{11} Q_{33} - Q_{13} Q_{31}}, r_{p s} = \frac{Q_{41} Q_{33} - Q_{43} Q_{31}}{Q_{11} Q_{33} - Q_{13} Q_{31}}, r_{s p} = \frac{Q_{11} Q_{23} - Q_{21} Q_{13}}{Q_{11} Q_{33} - Q_{13} Q_{31}} a n d r_{p p} = \frac{Q_{11} Q_{43} - Q_{41} Q_{13}}{Q_{11} Q_{33} - Q_{13} Q_{31}} .

Since the reflection coefficients are obtained, transverse shift (centroid shift perpendicular to the incident plane) of reflected light can be calculated as follows. Consider an incident Gaussian beam with a waist of w₀, its angular spectrum is

{\tilde{E}}_{i \pm} (k_{i x}, k_{i y}) = (e_{i x} + i σ e_{i x}) \frac{w_{0}}{\sqrt{2 π}} \exp [- \frac{w_{0}^{2} (k_{i x}^{2} + k_{i y}^{2})}{4}]

in which $σ = \pm 1$ corresponds to left-circularly and right-circularly polarized light, respectively. For the configuration which is shown in Fig. 1, the incident plane is xoz and the spin-dependent shift is along y-axis. Thus the angular spectrum of reflected light beam can be written as [37]

[\begin{matrix} {\tilde{E}}_{r}^{H} \\ {\tilde{E}}_{r}^{V} \end{matrix}] = [\begin{matrix} r_{p p} - \frac{k_{r y}}{k_{0}} (r_{p s} - r_{s p}) \cot θ_{i} & r_{p s} + \frac{k_{r y}}{k_{0}} (r_{p p} - r_{s s}) \cot θ_{i} \\ r_{s p} + \frac{k_{r y}}{k_{0}} (r_{p p} - r_{s s}) \cot θ_{i} & r_{s s} - \frac{k_{r y}}{k_{0}} (r_{p s} - r_{s p}) \cot θ_{i} \end{matrix}] [\begin{matrix} {\tilde{E}}_{i}^{H} \\ {\tilde{E}}_{i}^{V} \end{matrix}]

where

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

and

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

denote the horizontal and vertical components of angular spectra of reflected light beam,

k_{r y}

is y component of reflected wave vector. Here H and V represent horizontal and vertical polarizations, respectively. Combined with the condition of

\begin{array}{l} {\tilde{E}}_{r}^{H} = ({\tilde{E}}_{r +} + {\tilde{E}}_{r -}) / \sqrt{2} \\ {\tilde{E}}_{r}^{V} = i ({\tilde{E}}_{r -} - {\tilde{E}}_{r +}) / \sqrt{2} \end{array}

in which

{\tilde{E}}_{r +}

and

{\tilde{E}}_{r -}

denote the angular spectrum of the left and right circularly polarized components, then we can get the angular spectrum of LHCP and RHCP components of reflected beam, as well as their electric field distribution

E_{r} (x_{r}, y_{r}, z_{r}) = \iint {\tilde{E}}_{r} (k_{r x}, k_{r y}) \exp [i (k_{r x} x_{r} + k_{r y} y_{r} + k_{r z} z_{r})] d k_{r x} d k_{r y}

The spin-dependent splitting of reflected light can be defined as

δ_{V}^{\pm} = \frac{\iint E_{r \pm} \cdot E_{r \pm}^{*} y_{r} d x_{r} d y_{r}}{\iint E_{r \pm} \cdot E_{r \pm}^{*} d x_{r} d y_{r}}

The simplified expression is

δ_{V}^{\pm} = \frac{\pm k_{0} w_{0}^{2} \cot θ_{i} [\frac{| r_{p} |}{| r_{s} |} \cos (φ_{p} - φ_{s}) \mp | χ | \frac{| r_{p} |}{| r_{s} |} \sin (φ_{p} - φ_{p s}) - 1]}{\cot^{2} θ_{i} [\frac{{| r_{p} |}^{2}}{{| r_{s} |}^{2}} + \cos^{2} (φ_{s}) \mp 2 | χ | \sin (φ_{p s}) \cos (φ_{s})] + k_{0}^{2} w_{0}^{2} {(\cos (φ_{s}) \pm | χ | \sin (φ_{p s}))}^{2}}

in which

r_{p}, r_{s}, r_{p s}

are Fresnel reflection coefficients,

φ_{p}, φ_{s}, φ_{p s}

are their phase and

| χ | = | r_{p s} | / | r_{s} |

. It can be found from Eq. (24) that the difference between transverse shifts of left- and right-handed circularly polarized components is mainly reflects in items that contain the

| χ |

factor. If

| χ |

= 0,we can obtain

δ_{V}^{+}

= -

δ_{V}^{-}

, which means RHCP and LHCP components experience the same transverse shift and the reflected light splitting is symmetric. It can be presumed that in our structure SHEL with asymmetric spin-dependent splitting can be realized as the Kerr rotation of MO layer. Similarly, the calculation formula of transverse shift for PMOKE configuration can be simplified as

δ_{V}^{\pm} = \pm \frac{k_{0}^{2} w_{0}^{2} \cot θ_{i} [{| χ |}^{2} - \frac{| r_{s p} |}{| r_{s} |} [\sin (φ_{s p} - φ_{s}) \pm | χ | \cos (φ_{p s} - φ_{s p})] + \frac{| r_{p} |}{| r_{s} |} [\cos (φ_{p} - φ_{s}) \mp | χ | \sin (φ_{p} - φ_{p s})] - 1]}{k_{0}^{2} w_{0}^{2} (1 + {| χ |}^{2}) + \cot^{2} θ_{i} \frac{{| r_{p} |}^{2}}{{| r_{s} |}^{2}} \pm 2 k_{0}^{2} w_{0}^{2} | χ | \sin (φ_{p s} - φ_{s})}

For TMOKE configuration, the simplified calculation formula is

δ_{V}^{\pm} = \pm \frac{k_{0}^{2} w_{0}^{2} \cot θ_{i} [\frac{| r_{p} |}{| r_{s} |} \cos (φ_{p} - φ_{s}) - 1]}{k_{0}^{2} w_{0}^{2} + \cot^{2} θ_{i} \frac{{| r_{p} |}^{2}}{{| r_{s} |}^{2}}}

Obviously, reflection coefficients

r_{p s}

and

r_{s p}

are equal to zero in Eq. (26) because there is no Kerr rotation but a change of reflected light intensity in TMOKE configuration.

3. Results and discussion

In this section, we first focus on two kinds of structures including a magnetic film with DNG metamaterial substrate and another with SNG metamaterial ( $ε < 0$ in our simulation) substrate. At last, we make use of a realizable DNG metamaterial consists of silver nanostructures as the substrate to make our results convincing.

3.1 DNG metamaterial substrate

In order to study the influence of external magnetic field on photonic spin Hall effect of reflected light on the magnetic thin film with DNG metamaterial substrate, we calculated the Kerr rotation and spin-dependent splitting of reflected light beam in LMOKE, TMOKE and PMOKE configurations, respectively. In all cases the magnetic field is assumed to saturate the magnetization of the magnetic layer. Parameter in three cases are chosen as, $ε_{D N G} = - 2.996 + 0.997 i$ , $μ_{D N G} = - 1.49 + 0.86 i$ , $d_{Ce : Y I G} = 739 nm$ , the above three parameters are chose for a larger Kerr rotation and we will make use of a realizable DNG metamaterial based on silver nanostructures as substrate to verify our conclusion at the end of this paper. In Ce:YIG layer, $ε_{20} = 5.2347 + 0.0681 i$ and $ε_{21} = 0.029 - 0.0169 i$ [38]. We choose 1100 nm as operating wavelength because Ce:YIG shows a strong Faraday rotation at around 1 μm wavelength due to Ce³⁺-Fe³⁺ (tetrahedral) charge transfers [38,39].

(1) LMOKE configuration

Now we consider the first case, in which the substrate is DNG metamaterial and Ce:YIG layer is magnetized along x-direction (LMOKE configuration). When an incident light with vertical polarization is injected to the magnetic layer, the Kerr rotation of the reflected light is shown in Fig. 2(a). Due to enhancement of magneto-optical Kerr effect by DNG substrate [40], the Kerr rotation can reach near 45 degrees at a particular incident angle of about 37.2 degrees. More importantly, when the applied magnetic field is reversed, the direction of Kerr rotation is also reversed, while the absolute value of rotation angle remains unchanged. Figure 2(b) and 2(c) show the transverse shifts of LHCP and RHCP at upper surface of Ce:YIG layer when we change the direction of applied magnetic field. It can be seen that when the incident angle is around 37 degrees (also the region where Kerr rotation is larger in Fig. 2(a)), the transverse shift (the centroid shift perpendicular to the incident plane) of both LHCP and RHCP components increases obviously. This phenomenon can be explained by the influence of the polarization rotation on spin-dependent splitting in momentum space and position space [39]. On the other hand, the transverse shift of two circularly polarized components in Figs. 2(b) and 2(c) shows a significant asymmetry. In previous studies, the spin-dependent splitting mostly shows a symmetrical feature [2,4,5], that is, LHCP and RHCP components of reflected light shift in the opposite direction, but the distance is equal. Only individual studies have found asymmetric shifts, for example by changing the polarization characteristics of the incident light [41]. In Figs. 2(b) and 2(c), the spin-dependent splitting distances of two components are not equal in the most range of incident angle, and when the incident angle is about 37 degrees, LHCP and RHCP components of reflected light shift along the same direction. More interestingly, when the applied magnetic field is reversed, the splitting distance of two components will be interchanged while the direction of movement remains the same. This shows that the applied magnetic field can control the spin-dependent splitting of reflected light very well. The illustrations of Figs. 2(b) and 2(c) show the normalized electric field intensity distributions of the left and right-handed components of the reflected beam at the slice of x = 0, both of which exhibit a significant asymmetry in amplitude and centroid position, this phenomenon is similar to the results in Ref [41].

Fig. 2 (a) Kerr rotation of reflected light when applying two opposite magnetic fields to Ce:YIG layer along x axis. (b) and (c) Transverse shift (centroid shift perpendicular to the incident plane) of left- and right-circularly polarized light with vertical polarization incident for opposite magnetic fields ( + H and –H). (d) Asymmetry of SHE ( $S = δ_{V}^{+} + δ_{V}^{-}$ )and MOSHE shift ( $δ_{MO} = δ (- H) - δ (+ H)$ )of reflected light.

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In order to observe the asymmetry of the transverse shift of two circularly polarized components and influence of the applied magnetic field on spin-dependent splitting more intuitively, we define and calculate two parameters, including the asymmetry of SHE

S = δ_{V}^{+} + δ_{V}^{-}

and magneto-optical spin hall effect (MOSHE) shift

δ_{MO} = δ (- H) - δ (+ H)

in which

δ_{V}^{+}

and

δ_{V}^{-}

are spin-dependent shifts of LHCP and RHCP light respectively, as shown in Fig. 2(d). We can see that the value of S is larger than that at 37 degrees, because two spin components shift along the same direction at this incident angle, as shown in Figs. 2(b) and 2(c). On the other hand, we can also find that the sign of S is modulated by external magnetic field. MOSHE shift (

δ_{MO}

)of the left and right components are equal and numerically equal to the value of

| S |

, which is due to the size exchange between shifts of two components when external magnetic field is reversed.

(2) PMOKE configuration

Next we study the case in which the applied magnetic field is along z-axis. Figure 3(a) shows the Kerr rotation of the reflected light with positive ( + z) and negative (-z) magnetic field. Similar to LMOKE configuration, the direction of the magnetic field will directly affect the sign of Kerr rotation, which is consistent with the conclusion in Ref [42]. In contrast to LMOKE configuration, Kerr rotation of PMOKE configuration has two peaks, and the shift peaks of two circularly polarized components in Figs. 3(b) and 3(c) are well separated from each other. Moreover, it can be found from the illustration of Figs. 3(b) and 3(c) that the electric field intensity of two spin components also shows a more pronounced difference. Similarly, we calculated the asymmetry of PMOKE configuration, as shown in Fig. 3(d). Obviously, as the peaks of left and right spin components in Figs. 3(b) and 3(c) are separate there are two corresponding peaks of S parameter, too. The sign of S is also affected by the direction of magnetic field and $δ_{MO}$ is equal to $| S |$ .

Fig. 3 (a) Kerr rotation of reflected light when applying opposite magnetic field to Ce:YIG layer along z axis. (b) and (c) Transverse shift (centroid shift perpendicular to the incident plane) of left- and right-circularly polarized light with vertical polarization incident for opposite magnetic field ( + H and –H). (d) Asymmetry of SHEand MOSHE shift of reflected light.

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(3) TMOKE configuration

It is well known that the transverse magneto-optical Kerr effect is different from polar and longitudinal one. In LMOKE configuration and PMOKE configuration, when the incident light is reflected by the magnetic layer, the Kerr rotation occurs. From above results, we can found that the reflected light beam will undergo an asymmetric spin-dependent splitting and S is closely related to Kerr rotation. But in TMOKE configuration, the reflected light will get an additional nonreciprocal phase shift (NRPS) rather than a Kerr rotation [43]. It can be predicted that the transverse shift of two spin components of reflected light beam in case of TMOKE should be symmetrical. To verify this prediction, we calculate the intensity change of reflected light beam as well as spin-dependent splitting which are shown in Figs. 4(a), 4(b) and 4(c), respectively. Obviously, the transverse shifts of two spin components in reflected light is equal in size but opposite in direction, they are symmetric. In order to show the symmetric shift of two spin components more visually, normalized electric field intensity distribution of the reflected light at plane of x = 0 when the incident angle is 40 degrees is shown as a illustration in Fig. 4(b). It can be seen that the beam splitting of two components is symmetric and the electric field distribution is identical. Due to intensity change of reflected light in TMOKE configuration, MOSHE of light can also be observed as shown in Fig. 4(d).

Fig. 4 (a) and (b) Transverse shift (centroid shift perpendicular to the incident plane) of left- and right-circularly polarized light with vertical polarization incident for opposite magnetic fields ( + H and –H) along y axis. (c) Asymmetry of SHE ( $S = δ_{V}^{+} + δ_{V}^{-}$ ). (d) MOSHE shift ( $δ_{MO} = δ (- H) - δ (+ H)$ ) of reflected light.

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3.2 SNG metamaterial substrate(LMOKE)

In order to study the influence of the substrate material on spin-dependent splitting of the reflected light further, as a comparison, we replaced the DNG substrate with a single negative material substrate ( $ε < 0$ in our simulation). As shown in Fig. 5(a), with a single negative substrate the Kerr rotation of reflected light can still be enhanced, approaching near 30 degrees. The difference from a DNG substrate is that a large magnetic material thickness (about 5 times of the DNG substrate case) is required in the case of a SNG substrate. Therefore a DNG substrate is more conducive to integration for compact applications. This conclusion is consistent with the results of the terahertz band in [31]. Figures 5(b) and 5(c) shows transverse shifts of LHCP and RHCP, two spin components exhibit a significant asymmetric shift, and the electric field amplitudes are not equal. The asymmetry S is shown in Fig. 5(d).

Fig. 5 (a) Kerr rotation of reflected light when applying two opposite magnetic fields to Ce:YIG layer along x axis. (b) and (c) Transverse shift (centroid shift perpendicular to the incident plane) of left- and right-circularly polarized light with vertical polarization incident for opposite magnetic fields ( + H and –H). (d) Asymmetry of SHE ( $S = δ_{V}^{+} + δ_{V}^{-}$ ) of reflected light.

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3.3 Substrate with realizable DNG metamaterial

To increase the persuasiveness of above results, we use the low loss double negative metamaterial based on silver nanostructures fabricated by Dolling et al. in Ref [33] to study the effect of magnetic field on SHEL in LMOKE configuration. The structure of this DNG metamaterial is a combination of double-plate pairs and long wires of silver which can be seen as “magnetic atoms” and “electric atoms” respectively [33]. The wavelength of the incident light in our simulation is 1410 nm. According to the experimental results in Ref [33], the dielectric constant and the permeability of DNG are $ε_{3} = - 1.5 + 0.2 i$ and $μ_{3} = - 0.8 + 0.6 i$ . By adjusting the thickness of the magnetic layer, we get a Kerr rotation of approximately 10 degrees, as shown in Fig. 6(a). The SHE shift of reflected light with positive and negative magnetic field is shown in Figs. 6(b) and 6(c). It can be seen that the modulation characteristic of the magnetic field to SHEL is similar with that in Fig. 2 since the Kerr rotation is relatively small in this structure. The value of S is about 1.2 microns, it is still obvious. This fully illustrates the practical feasibility of modulating SHEL using magnetic field.

Fig. 6 (a) Kerr rotation of reflected light when applying two opposite magnetic fields to Ce:YIG layer along x axis. (b) and (c) Transverse shift (centroid shift perpendicular to the incident plane) of left- and right-circularly polarized light with vertical polarization incident for opposite magnetic fields ( + H and –H). (d) Asymmetry of SHE ( $S = δ_{V}^{+} + δ_{V}^{-}$ ) of reflected light.

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4. Conclusion

In this paper, we study SHE of reflected light on a dielectric magneto-optical thin film of Ce₁Y₂Fe₅O₁₂ (Ce:YIG) with a double-negative metamaterial substrate. The transverse shifts (centroid shifts of the left- and the right-handed circularly polarized light which perpendicular to the incident plane) of reflected light beam in three types of MOKE configurations are obtained and discussed, which show different features. In addition, we define and calculate the parameters S and $δ_{MO}$ to characterize the influence of external magnetic field on spin-dependent splitting. A maximum spin-dependent splitting between LHCP and RHCP components about 9.2 μm and a magneto-optical spin Hall effect (MOSHE) shift of 9 μm are observed (in PMOKE configuration). Based on simulation results, the influences of magnetic field direction and substrate material on transverse shifts of LHCP and RHCP light are analyzed. In order to make our results more convincing, we use a DNG metamaterial based on silver nanostructures as substrate to verify our conclusion. The simulation results in this paper provide a new way to manipulate spin Hall effect of light flexibly.

Funding

National Natural Science Foundation of China (NSFC) (61505016, 61475031, 51522204 and 11674234); Project of Sichuan Provincial Department of Education (15ZA0183); Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1402907); Science and Technology Bureau of Chengdu (2015-HM01-00579-SF); Scientific research fund of Chengdu University of Information Technology (J201417).

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Spin Hall effect of reflected light in dielectric magneto-optical thin film with a double-negative metamaterial substrate

Abstract

Corrections

1. Introduction

2. Theoretical analysis

3. Results and discussion

3.1 DNG metamaterial substrate

(1) LMOKE configuration

(2) PMOKE configuration

(3) TMOKE configuration

3.2 SNG metamaterial substrate(LMOKE)

3.3 Substrate with realizable DNG metamaterial

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (16)

Optics Express