Chirality induced asymmetric spin splitting of light beams reflected from an air-chiral interface

Mengjiang Jiang; Hai Lin; Linqing Zhuo; Wenguo Zhu; Heyuan Guan; Jianhui Yu; Huihui Lu; Jieyuan Tan; Zhe Chen

doi:10.1364/OE.26.006593

1. Introduction

For a bounded light beam, the reflection and refraction cannot be exactly described by the geometrical optics (Snell’s law and Fresnel formulas) alone. The reflected/refracted beam will undergo shifts in directions parallel and perpendicular to the incident plane, which are so-called Goos-Hänchen (GH) and Imbert–Fedorov (IF) shifts, respectively [1–4]. The later one is spin dependent and also known as spin Hall effect of light, since it results from the spin-orbit interaction, governed by the conservation law of angular momentum [5–7]. Therefore, for a linearly polarized incident beam, the right-handed and left-handed circular polarization (RCP and LCP) components of the reflected/refracted beam will shift toward opposite directions and split spatially [8]. Weak measurement technology is used in its observation, since the spin splitting is general smaller than a wavelength [1,8].

Recently, much attention has been paid to the enhancement of spin splitting of light beam due to its potential applications in precision metrology, such as identifying the layer of graphene [9] and measuring the concentration of glucose and fructose [10]. It was demonstrated that the spin splitting of the reflected beam can be enhanced by launching a Gaussian beam near the Brewster angle [11]. Taking advantage of surface plasmon resonance, a spin dependent displacement up to 11.5 μm has been obtained theoretically [12,13]. To enhance the spin-orbital interaction, researchers transmitted light beams through thin metamaterial slabs, and the spin-dependent displacement can reach a few tens of wavelength [14–17]. Very recently, a tunable spin-dependent displacement of light beam was demonstrated by transmitting higher-order Laguerre-Gaussian (LG) beams through graphene metamaterials [18].

A chiral metamaterial has recently drawn significant attention due to its strong chiral behaviors such as optical activity and circular dichroism [19].Through pragmatic designs, the chiroptical response of chiral metamaterials can be several orders of magnitude higher than that of natural chiral media [20].Meanwhile, chiral metamaterials provide a simpler pathway to realize negative refraction [21,22]. Chiral metamaterials with negative refractive index based on structures of split-ring resonators [23] and four U-shaped resonators have been reported successively [24]. The unique properties of chiral metamaterial offer special opportunity to tailor the spin Hall effect of light. In 2011, the transverse shifts of Gaussian beams reflected from an air-chiral interface were investigated [19]. It illustrated that the transverse shifts can be tuned by the chiral parameters [25].Lately, asymmetric spin splitting was predicted theoretically when Gaussian beams reflected by a thin chiral metamaterial slab [26].

In this paper, we investigate systematically the asymmetric spin splitting of Gaussian beams reflected from an air-chiral interface. We find that both the displacements and energies of the two spin components of the reflected beam are different under asymmetric spin splitting. And giant asymmetric spin splitting can be obtained near points of |r_pp| = |r_sp|, where r_pp and r_sp are the Fresnel reflection coefficients. We further study the asymmetric spin splitting of light beams carrying orbital angular momentum (OAM).The OAM will induce two additional terms in centroid displacements of RCP and LCP components of the reflected beam. These two terms alone will lead to asymmetric spin splitting. The corporation effect of the asymmetric spin splitting originated from Gaussian envelop and OAM can increase the asymmetric spin splitting effect of reflected beam.

2. Theory and model

We consider a linearly polarized beam reflected from an air-chiral interface with an incident angle of θ, as shown in Fig. 1. The global coordinate system is (x_g, y_g, z_g), while (x_i, y_i, z_i) and (x_r, y_r, z_r) are coordinate systems of the incident and reflected beams, respectively. With the help of the boundary conditions, we can obtain a 2 × 2 Fresnel reflection matrix describing the relations between the incident and reflected plane waves at the air-chiral interface [27]:

r = [\begin{matrix} r_{p p} & r_{p s} \\ r_{s p} & r_{s s} \end{matrix}],

where

r_{p p} = \frac{(1 - g^{2}) (\cos θ_{+} + \cos θ_{-}) \cos θ - 2 g (\cos^{2} θ - \cos θ_{+} \cos θ_{-})}{(1 + g^{2}) (\cos θ_{+} + \cos θ_{-}) \cos θ + 2 g (\cos^{2} θ + \cos θ_{+} \cos θ_{-})},

r_{s p} = r_{p s} = \frac{- 2 i g (\cos θ_{+} - \cos θ_{-}) \cos θ}{(1 + g^{2}) (\cos θ_{+} + \cos θ_{-}) \cos θ + 2 g (\cos^{2} θ + \cos θ_{+} \cos θ_{-})},

r_{s s} = \frac{(1 - g^{2}) (\cos θ_{+} + \cos θ_{-}) \cos θ + 2 g (\cos^{2} θ - \cos θ_{+} \cos θ_{-})}{(1 + g^{2}) (\cos θ_{+} + \cos θ_{-}) \cos θ + 2 g (\cos^{2} θ + \cos θ_{+} \cos θ_{-})},

With g = (ε/μ)^1/2, cos θ _± = (1-sin²θ/n² _± )^1/2, where θ _± and n _± = n ± к are, respectively, the refracted angles and the refractive indices of RCP and LCP components; ε, µ, and к are relative permittivity, permeability and the chirality parameters of the chiral metamaterial, respectively; n = (εμ)^1/2being the average refraction index for the two spin components of chiral metamaterial. Since the refraction index for RCP and LCP components in chiral metamaterial are differently, their behavior at chiral metamaterial will be different accordingly.

Fig. 1 Illustration of the asymmetric spin splitting of light beam reflected from an air-chiral interface. For a horizontal incident polarization, the two opposite spin components of the reflected beam will undergo asymmetric displacements along y_r axis.

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To accurately describe reflection of a bounded light beam, we need to determine the reflection of arbitrary wave-vector components, since a bound light beam can be considered as a superposition of plane waves. After the coordinate rotation, and combining with Eq. (1), we can obtain the reflected angular spectrum by means of the relation Ẽ_r(k_rx, k_ry) = MẼ_i(k_ix, k_iy). Here, Ẽ_r and Ẽ_i are the angular spectrums of the reflected and incident beams, k_rx = -k_ix, k_ry = k_iy are the transverse wavenumber, respectively. And the matrix M can be expressed as [2,28]

[\begin{matrix} r_{p p} & r_{p s} + M k_{r y} / k_{0} \\ r_{s p} - M k_{r y} / k_{0} & r_{s s} \end{matrix}],

where M = (r_pp + r_ss) cotθ, k₀ = 2π/λ is the wavenumber in vacuum with λ being the wavelength in free space. Owing to the tiny in-plane spread of wave vectors, the Fresnel coefficients should be expanded into Taylor series around the central wave vector. By making the first order approximation, we have [2,19]

r_{a b} (k_{i x} / k_{0}) = r_{a b} (0) + k_{i x} / k_{0} r_{a b}^{'} (0),

with a and b stand for either s or p.

r_{a b}^{'}

denotes the derivative of Fresnel reflection coefficients with respect to the incident angle θ [19].

In order to study the influence of OAM on the spin splitting, LG beams are chosen as incident beams. The angular spectra of incident LG profiles are ${\tilde{ϕ}}_{l} \propto {[(- i k_{i x} + s_{l} k_{i y})]}^{| l |} \exp [- (k_{i x}^{2} + k_{i y}^{2}) w_{0}^{2} / 4],$ where ℓ and w₀ are the incident OAM and the radius of beam waist; and s_ℓ = sign (ℓ) being the sign of ℓ. For a horizontal (H) incident polarization, i.e., ${\tilde{E}}_{i}^{} = {\tilde{ϕ}}_{l} {\hat{e}}_{i x}$ , the angular spectrum of the reflected beam can be obtained according to Eqs. (1-4). In circular polarization basis, ${\hat{e}}_{r \pm} = [{\hat{e}}_{r x} \pm i {\hat{e}}_{r y}] / 2^{1 / 2}$ , the RCP and LCP reflected components are:

{\tilde{E}}_{r}^{\pm} = {[r_{p p} - \frac{k_{r x} r_{p p}^{'}}{k_{0}}] \mp i [r_{s p} - \frac{k_{r x} r_{s p}^{'}}{k_{0}} - \frac{M k_{r y}}{k_{0}}]} {\tilde{ϕ}}_{l} {\hat{e}}_{r \pm} .

The centroids of the RCP and LCP components of the reflected beam along y_r axis are defined as

Δ_{\pm} = 〈 {\tilde{E}}_{r}^{\pm} | i \partial_{k_{r y}} | {\tilde{E}}_{r}^{\pm} 〉 / 〈 {\tilde{E}}_{r}^{\pm} | {\tilde{E}}_{r}^{\pm} 〉

[1]. After some straightforward calculation, we arrive at

Δ_{\pm} = {Im [r_{s p}^{*} M] \pm Re [r_{p p}^{*} M] - l Re (r_{p p}^{*} r_{p p}^{'} + r_{s p}^{*} r_{s p}^{'}) \pm l Im (- r_{p p}^{*} r_{s p}^{'} + r_{s p}^{*} r_{p p}^{'})} / W_{\pm} k_{0},

where the energies of the RCP and LCP components after reflected from the air-chiral interface are given respectively

\begin{matrix} W_{\pm} = {| r_{p p} |^{2} + | r_{s p} |^{2} \pm 2 Im [r_{p p}^{*} r_{s p}] + (| l | + 1) {| r_{p p}^{'} |^{2} + | r_{s p}^{'} |^{2} \\ \pm 2 Im [r_{p p}^{'}^{*} r_{s p}^{'}] + | M |^{2}} / k_{0}^{2} w_{0}^{2}} . \end{matrix}

Noting that the first two terms of Eq. (6) are OAM-independent, resulting from the Gaussian envelop, while the last two terms are OAM-dependent, originating from vortex structure. The second and fourth terms are spin-dependent, thus the reflected RCP and LCP components will be shifted toward opposite directions. The first and third terms will shift the two opposite spin components together. Therefore, for a Gaussian incident beam, the cooperation effect of the first two terms (the last two terms vanish) will lead to an asymmetric spin splitting. For LG incident beams, two additional OAM-dependent terms emerge. These two terms alone will also cause asymmetric spin splitting, which provides an efficient method to enhance the asymmetric spin splitting effect of the reflected beam.

For an isotropic material, the nondiagonal elements of Fresnel coefficient r_sp and r_ps vanish, and the reflected Gaussian beam will undergo symmetric spin splitting. According to Eq. (6)., the asymmetric spin splitting will occur for optical systems with nonzero nondiagonal elements of Fresnel coefficient. Such optical systems include air-chiral interface [19], air-topological insulator interface [2], and graphene–substrate system with an imposed static magnetic field [28,29]. The air-chiral interface shows advantage since giant asymmetric spin splitting can be obtained as will be shown below.

3. Result and discussion

To study the spin Hall effect of light at an interface between air and chiral metamaterial, we calculated the transverse displacement of a reflected beam. The results of the reflectivity, the spin-dependent displacement along y_raxis and the energy ratio of the reflected RCP and LCP components as functions of the incident angle θ are shown in Fig. 2, respectively. Here, R = 10lg (W₊/W_-) is introduced to describe the energy ratio between the reflected RCP and LCP components. Different cases of к = 0, 0.1, and 0.49 are given in the first, second, and third columns, respectively. The other parameters are set to be n = 1.414, and w₀ = 100λ, respectively.

Fig. 2 Dependences of the reflectivity (the first row), the spin-dependent displacements (the second row), and the energy ratio of the reflected RCP and LCP components (the third row) on θ for the cases of к = 0 (the first column), 0.1 (the second column), and 0.49 (the third column), respectively. In our calculations, n = 1.414 and w₀ = 100λ.

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The numerical results show that for an achiral medium (к = 0), the cross-polarized |r_sp| vanishes, and the energy ratio between the reflected RCP and LCP components R = 0 dB. That is the two opposite spin components of the reflected beam will shift toward opposite directions with identical magnitude, resulting in a symmetric spin splitting. Near the valley of |r_pp|, the spin-dependent displacements |Δ _± | can reach 20.3λ. However, for a chiral medium (к≠0), the cross-polarized |r_sp|≠0. When к = 0.1, there are two intersection points between |r_pp| and |r_sp|, i.e., θ = 53.21° and 56.95°, as shown in Fig. 2(b1). At the first point of |r_pp| = |r_sp| (θ = 53.21°), the energy ratio between the reflected RCP and LCP components is R = 30 dB. The energy of the reflected LCP component almost vanishes, and the reflected beam is nearly in RCP state. In contrast, at the second point of |r_pp| = |r_sp| (θ = 56.95°), the reflected beam is almost in LCP state. Around the first (second) point of |r_pp| = |r_sp|, the displacement of the reflected LCP Δ_- (RCP Δ₊) component will vary rapidly with the incident angle, while the displacement of the other spin component keeps, thus the reflected beam undergoes asymmetrical spin splitting. When к = 0.49, there is only one intersection point between |r_pp| and |r_sp|, as shown in Fig. (c1). The displacements of the LCP component can reach 25.7λ, while the displacement of the RCP component is smaller than 0.1λ.

Fora chiral metamaterial, the responses for LCP and RCP components are different due to the intrinsic chiral asymmetry of the structure. Figure 3(a) and (b) show the changes of the spin-dependent displacements of the reflected beam with chirality parameter к. In the calculation, θ = 54°, n = 1.414.Obviously, the profiles of the displacements of the reflected RCP (solid lines) and LCP (dashes lines) components are on the center symmetry of the origin point. As shown in Fig. 3(a), the displacement of the reflected RCP ∆₊ component will change rapidly from −21.04λ to 20.91λ, when the chirality parameter changes from к = −0.050 to −0.038. The displacement of the reflected LCP Δ_- component has similar change near point of к = 0.044. Therefore, we conclude that spin-dependent displacements are sensitive to the chirality parameter, thus have the potential application in metrological measurement for chiral medium. Noting that there are two peaks around к = ± 0.604, respectively. The reason can be explained as follows: in the case of к = 0.604, the incident angle θ = 54° is exactly the critical angle for total reflection of LCP component. According to Eqs. (2a), (2b), and (2c), the reflection coefficients satisfy relationship of ir_pp = r_sp, and |r_ss| is much smaller than |r_pp| and |r_sp|. Therefore, one spin component of the reflected beam will undergo relative large shift near к = 0.604, according to Eq. (6). Similarly, the RCP component will undergo relative large shift around к = −0.604.One can conclude that the reflected beam will undergo giant asymmetric spin splitting near points of |r_pp| = |r_sp| for H incident polarization. Similar asymmetric spin splitting also occurs for vertical incident polarization near points of |r_ss| = |r_ps|.

Fig. 3 Dependences of the spin-dependent displacements∆ _± on chirality parameter к for (a) к = −1~1and (b)к = −0.1~0.1, respectively. In the calculation, θ = 54°, n = 1.414.

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The incident beam waist is another important parameter that affects the spin-dependent displacements of the reflected beam. Fig. 3(b) gives the spin-dependent displacements of the reflected beams for different incident beam waists. As shown in the figure, the peak values of spin-dependent displacements of reflected beam will increase with the incident beam waistw₀. The peak displacement can be up to 64.7λ when w₀ = 300λ. However, away from the peak positions, the spin-dependent displacements are identical for all the cases of w₀ = 100λ, 200λ, and 300λ.

The shape of the incident beam will affect the spin-dependent displacements of the reflected beam. As shown in Eq. (6), there are two additional ℓ-dependent terms for the incident beam carrying OAM. The displacements of the reflected RCP and LCP components for the incident OAM of ℓ = −2:1:2 are demonstrated in Fig. 4, where the spin-dependent displacements ∆ _± and the energy ratio between the reflected RCP and LCP components R are shown as functions of the incident angle θ for к = 0.1 (first column) and of the chirality parameter к for θ = 54° (second column), respectively. The energy ratios |R| are independent of ℓ, excepting near their peak positions. The total spin-dependent displacements of the reflected beam are resulted from the cooperation effect of the ℓ-independent andℓ-dependent displacements, which are stemmed from Gaussian envelop and vortex structure, respectively. The ℓ-dependent displacements change signs when the incident OAM varies from positive to negative values. Therefore, the ℓ-independent and ℓ-dependent displacements can be of identical or opposite in sign. The displacement directions of both the reflected RCP and LCP components ∆ _± are determined by the sign of ℓ-dependent displacements, since ℓ-dependent displacements are larger than ℓ-independent displacements even for ℓ = ± 1. And |∆ _± | increases with ℓ for higher-order LG incident beams. When ℓ = −2, the displacement of the reflected LCP component can reach 64.5λ, which is three times larger than the displacement of LCP component for foundational Gaussian beam (ℓ = 0).

Fig. 4 The spin-dependent displacements∆ _± (the first and second rows) and the energy ratio between the reflected RCP and LCP components R(the third row) as functions of the incident angle θ for к = 0.1(first column), and of the chirality parameter к for θ = 54°(second column), respectively.

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The spin-dependent displacement of the reflected beam as a function of the average refractive index for the two spin components of chiral metamaterial is shown in Fig. 5. In the calculation, θ = 50.79° and к = 0.58. The red and blue lines correspond to the RCP and LCP components, respectively. As shown in Fig. 5, the displacement of the reflected RCP component changes smoothly with n, while the displacement of reflected LCP component experiences two sudden change near the positions of n = 1.355 and 1.417. When n = 1.417, the reflection coefficients satisfy |r_pp| = |r_sp|, where large asymmetric spin-dependent displacement occurs as discussed above. The sudden change near position of n = 1.355 is because the chosen incident angle θ = 50.79° is equal to the critical angle for total reflection, similar to the case of Fig. 3(a).

Fig. 5 The spin-dependent displacements of RCP (red color) and LCP (blue color) components of the reflected beams changing with the average refractive index of the chiral metamaterial n. In our calculations, θ = 50.79°, к = 0.58, and w₀ = 100λ.

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4. Conclusions

We have shown the chirality induced asymmetric spin splitting for foundational Gaussian and higher-order LG beams reflected from an air-chiral interface. For light beam carrying OAM, the asymmetric spin splitting contains two parts originated from Gaussian envelop (ℓ-independent) and vortex structure (ℓ-dependent).The asymmetric spin splitting can be enhanced significantly near the points of |r_pp| = |r_sp|.One of the spin component of the reflected beam can undergo a displacement up to 64.5λfor OAM ℓ = −2, while the displacement of the other component is 43.2λ. This asymmetric spin splitting can be well controlled by chiral parameter and the incident OAM. Moreover, the spin-dependent displacements are sensitivity to the parameter of chiral metamaterial (к and n), thus has potential application in its metrological measurement [9,10].

Funding

National Natural Science Foundation of China (NSFC) (61705086, 61505069, 61675092); Guangzhou Science and Technology Program key projects (2017A030313375, 2017A010102006); Science and Technology Project of Guangzhou (201605030002, 20160404005).

References and links

1. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013). [CrossRef]

2. X. Zhou, J. Zhang, X. Ling, S. Chen, H. Luo, and S. Wen, “Photonic spin Hall effect in topological insulators,” Phys. Rev. A 88(5), 053840 (2013). [CrossRef]

3. X. Zhou and X. Ling, “Unveiling the photonic spin Hall effect with asymmetric spin-dependent splitting,” Opt. Express 24(3), 3025–3036 (2016). [CrossRef] [PubMed]

4. S. Grosche, M. Ornigotti, and A. Szameit, “Goos-Hänchen and Imbert-Fedorov shifts for Gaussian beams impinging on graphene-coated surfaces,” Opt. Express 23(23), 30195–30203 (2015). [CrossRef] [PubMed]

5. K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96(7), 073903 (2006). [CrossRef] [PubMed]

6. X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light Sci. Appl. 4(5), e290 (2015). [CrossRef]

7. X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017). [CrossRef] [PubMed]

8. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008). [CrossRef] [PubMed]

9. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012). [CrossRef]

10. X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032–4035 (2016). [CrossRef] [PubMed]

11. H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin Hall effect of light near the Brewster angle on reflection,” Phys. Rev. A 84(4), 043806 (2011). [CrossRef]

12. Y. Xiang, X. Jiang, Q. You, J. Guo, and X. Dai, “Enhanced spin Hall effect of reflected light with guided-wave surface plasmon resonance,” Photon. Res. 5(5), 467–472 (2017). [CrossRef]

13. W. Zhu, L. Zhuo, M. Jiang, H. Guan, J. Yu, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Controllable symmetric and asymmetric spin splitting of Laguerre-Gaussian beams assisted by surface plasmon resonance,” Opt. Lett. 42(23), 4869–4872 (2017). [CrossRef] [PubMed]

14. T. Tang, J. Li, Y. Zhang, C. Li, and L. Luo, “Spin Hall effect of transmitted light in a three-layer waveguide with lossy epsilon-near-zero metamaterial,” Opt. Express 24(24), 28113–28121 (2016). [CrossRef] [PubMed]

15. M. Jiang, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Giant spin splitting induced by orbital angular momentum in an epsilon-near-zero metamaterial slab,” Opt. Lett. 42(17), 3259–3262 (2017). [CrossRef] [PubMed]

16. W. Zhu and W. She, “Enhanced spin Hall effect of transmitted light through a thin epsilon-near-zero slab,” Opt. Lett. 40(13), 2961–2964 (2015). [CrossRef] [PubMed]

17. T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6(1), 30762 (2016). [CrossRef] [PubMed]

18. W. Zhu, M. Jiang, H. Guan, J. Yu, H. Lu, J. Zhang, and Z. Chen, “Tunable spin splitting of Laguerre–Gaussian beams in graphene metamaterials,” Photon. Res. 5(6), 684–688 (2017). [CrossRef]

19. G. Xu, T. Zang, H. Mao, and T. Pan, “Transverse shifts of a reflected light beam from the air-chiral interface,” Phys. Rev. A 83(5), 053828 (2011). [CrossRef]

20. Z. Wang, F. Cheng, T. Winsor, and Y. Liu, “Optical chiral metamaterials: A review of the fundamentals, fabrication methods and applications,” Nanotechnology 27(41), 412001 (2016). [CrossRef] [PubMed]

21. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metametrial with Negative Index due to Chirality,” Phys. Rev. B 79(3), 035407 (2009). [CrossRef]

22. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. 17(5), 695–706 (2003). [CrossRef]

23. B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009). [CrossRef]

24. Z. F. Li, R. K. Zhao, T. Koschny, M. Kafesaki, K. B. Alici, E. Colak, H. Caglayan, E. Ozbay, and C. M. Soukoulis, “Chiral metamaterials with negative refractive index based on four “U” split ring resonators,” Appl. Phys. Lett. 97(8), 081901 (2010). [CrossRef]

25. H. Wang and X. Zhang, “Unusual spin Hall effect of a light beam in chiral metamaterials,” Phys. Rev. A 83(5), 053820 (2011). [CrossRef]

26. Y. Y. Huang, Z. W. Yu, and L. Gao, “Tunable spin-dependent splitting of light beam in a chiral metamaterial slab,” J. Opt. 16(7), 075103 (2014). [CrossRef]

27. J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5(4), 417–443 (1996). [CrossRef]

28. L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017). [CrossRef]

29. T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magneto-Optical Modulation of Photonic Spin Hall Effect of Graphene in Terahertz Region,” Adv. Opt. Mater 6(4), 201701212 (2018).

Chirality induced asymmetric spin splitting of light beams reflected from an air-chiral interface

Abstract

1. Introduction

2. Theory and model

3. Result and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (5)

Equations (9)

Optics Express