Abstract

We show theoretically that after transmitted through a thin anisotropic ε-near-zero metamaterial, a linearly polarized Gaussian beam can undergo both transverse spatial and angular spin splitting. The upper limits of spatial and angular spin splitting are found to be the beam waist and divergence angle of incident Gaussian beam, respectively. The spin splitting of transmitted beam after propagating a distance z depends on both the spatial and angular spin splitting. By combining the spatial and angular spin splitting properly, we can maximize the spin splitting of propagated beam, which is nearly equal to the spot size of Gaussian beam w(z).

© 2017 Optical Society of America

1. Introduction

Reflection and transmission of plane waves at an interface of two isotropic materials are well described by Snell’s law and Fresnel formulae, which however are not accurately complied by bounded light beams [1–6]. Owing to the diffractive corrections, the gravity centers of bounded beams will undergo small displacements in directions parallel and perpendicular to the plane of incidence [1]. The later one is the so called Imbert–Fedorov (IF) shifts [1], which results from the spin-orbital interaction and governed by the conservation law of angular momentum [1,7]. The IF shift is spin dependent. Therefore, for a linearly polarized incident beam, two opposite spin components of reflected/transmitted beam will shift toward opposite directions, thus split spatially [8,9]. In general, the spin splitting is limited by a fraction of the wavelength [1]. However, researchers found that the spatial spin splitting of reflected light can be enhanced by launching the incident beam near the Brewster angle [10,11]. Using the eigenpolarization concept, Götte and associates shown that the IF shift at Brewster incidence is very similar in form and magnitude to the Goos-Hänchen shift at critical incidence [12]. By choosing the eigenpolarization appropriately, a splitting up to 10 wavelengths has been obtained experimentally [12]. In 2015, Ren, et. al. demonstrated a resolvable spin separation for one dimensional (1D) Gaussian beam with a certain beam waist at an air-glass interface [13]. However, this result cannot be extended to two dimensional (2D) Gaussian beam. Different from reflected beam, the spin splitting of transmitted beam at near-Brewster incidence is tiny [14,15]. It is found recently that the spin splitting can be enhanced into a few tens of wavelengths when a Gaussian beam is transmitted through a hyperbolic metamaterial waveguide [16] or a thin epsilon-near-zero slab [17]. In addition to the spatial spin splitting, the reflected/transmitted beam can undergo angular spin splitting. The angular spin splitting is tiny, and has been observed by weak measurement technique [18,19].

Here, we focus our attention on both spatial and angular spin splitting of transmitted beam through a thin anisotropic ε-near-zero metamaterial, which is constituted by metallic nanorod arrays [20]. Assume that the permittivity of metamaterial for the electric field perpendicular to the nanorod axes ε// is identical to the permittivity of surrounding medium, the Fresnel transmission coefficient for s-wave is equal to 1 for arbitrary incident angle. If the perpendicular component of the permittivity ε is near zero, both the amplitude and phase of transmission coefficient for p-wave vary with incident angle θi and thickness of metamaterial d, which results in the spatial and angular spin splitting. By controlling θi and d, we can optimize the spin splitting of propagated beam after transmitted through the metamaterial. The upper limit of spin splitting at z plane is found be equal to the radius of the spot size of Gaussian beam at that plane w(z).

2. Theory and model

Let’s consider an anisotropic metamaterial formed by embedding arrays of parallel metallic wires in a dielectric matrix. The effective dielectric tensor of the anisotropic metamaterial has the form: [ε//, ε//, ε], where ε// is determined by the permittivity of background dielectric, while ε can be close to zero for the incident wave with a frequency around effective plasma frequency of metamaterial [20,21]. We assume that the background medium of metamaterial and the surrounding medium are the same, which are set to be air (ε// = 1) for simplicity. And ε is chosen to be a positive but near-zero value. Now, a monochromatic Gaussian beam is launched onto the anisotropic metamaterial with an incident angle of θi. As shown in Fig. 1, the metamaterial is placed at z = 0 plane, which coincides with the plane of beam waist. The incident beam will transmit partly through metamaterial. The local coordinate systems attached to the incident and transmitted beams are identical, (x,y,z). For a x-polarized incident Gaussian beam, the angular spectrum is E˜i=w0/(2π)1/2exp[(kx2+ky2)w02/4]e^x. According to Ref [1,17], the transmitted angular spectrum is given by

E˜t=w02πexp[(kx2+ky2)w024]{[tpθi+kxk0tpθi]e^x+[kyk0(tpθitsθi)cotθi]e^y},
where k0 is wavenumber in free space; tpθi and tsθi are the Fresnel transmission coefficients for p- and s- waves incident at θi; ∂tp/θi is the first-order derivative of transmission coefficient. For a thin anisotropic metamaterial, the Fresnel transmission coefficients are in form of
tpθi=1cosδpisinδp2ε(1+cos2θi)sin2θicosθiε//ε(εsin2θi),
tsθi=1cosδsisinδs2cos2θi+ε//cosθiε//sin2θi,
where δp = k0d(ε//-ε//sin2θi/ε)1/2, δs = k0d(ε//-sin2θi)1/2 with d being the thickness of metamaterial. When ε// = 1, tsθi = exp[ik0d(1-sin2θi)1/2], as if the s-plane wave is propagating in the free space. However, the transmission property of p-wave varies with ε. The difference between the transmission coefficients for p- and s-waves will result in the spatial and angular spin splitting [17]. From the angular spectrum Eq. (1), the complex amplitude of transmitted beam can be calculated easily. In circular polarization basis ê ± = êx ± iêy, the transmitted field is
E±=1πw0z0z0+izexp[k0(x2+y2)2(z0+iz)][tpθi+ixz0+iztpθi±y(tpθitsθi)cotθiz0+iz]e^±,
where E ± are the right and left handed circular polarization (RCP and LCP) components; z0 = k0w02/2 is the Rayleigh length. At a given plane z = const, the transverse displacements of two opposite spin (RCP and LCP) components of the transmitted beam compared to the geometrical optics prediction are defined as [10]
Δ±=y|E±|2dxdy/|E±|2dxdy.
The displacements can be written as a combination of z-dependent and z-independent terms: Δ ± = ΔY ± + zΔθ ± [22]. The former corresponds to transverse spatial shifts, while the latter can be regarded as transverse angular shifts. By substituting Eqs. (3) into (4), we have
ΔY±=±w0|tpθi||M|cosφ|tpθi|2+1k02w02|tpθi|2+|M|2,
Δθ±=±2k0w0|tpθi||M|sinφ|tpθi|2+1k02w02|tpθi|2+|M|2,
where M=(tpθitsθi)cotθi/k0w0, φ=arg[tpθi*M] with arg(x) being the argument of complex x. From Eqs. (5) and (6) one finds that, the spatial and angular shifts of two opposite spin components of transmitted beam are the same in magnitude, but opposite in signs. The spatial and angular spin splitting are ΔY = ΔY+Y- and Δθ = Δθ+θ-, respectively. Generally, the transmitted beam can have spatial and angular spin splitting simultaneously. When phase parameter φ = ± π/2, ΔY = 0, the transmitted beam only undergoes angular spin splitting. While φ = 0 or π, Δθ = 0, the transmitted beam splits spatially. In the paraxial region, |tpθi|>>|tp/θi|/k0w0, so the term |tp/θi|2/k02w02 in the denominator of Eq. (5) can be neglected. Therefore, the maximum spatial spin splitting is approximately obtained when the conditions: 1) |tpθi|2=|M|2, and 2) φ = 0 or π are satisfied simultaneously. These two conditions ensure |tpθi||M|/[|tpθi|2+|M|2] and |cosφ| reaching their maximum values, respectively. Under these conditions, the spatial spin splitting of transmitted beam is given by
|ΔY|m=w01+12|tpθi|2k02w02|tpθi|2.
Since |tpθi|>>|tp/θi|/k0w0, |ΔY|m approaches closely to w0 but cannot exceed it. Therefore, the incident beam waist is the upper limit of the spatial spin splitting. For the maximum angular spin splitting, condition (1) is still required, while condition (2) is changed to φ = ± π/2. Under these two conditions, the angular spin splitting is
|Δθ|m=2k0w011+12|tpθi|2k02w02|tpθi|2.
Similarly, the upper limit of spin angular splitting is the divergent angle of incident Gaussian beam θd = 2/k0w0. It is worth noted that, the two conditions for the maximum spatial/angular spin splitting might not be achieved simultaneously for a certain incident beam waist. In the following, we will try to approach the upper limits of spatial and angular spin splitting by adjusting the thickness of metamaterial and incident angle.

 figure: Fig. 1

Fig. 1 The schematic of spin splitting of transmitted beam. A linearly polarized Gaussian beam is incident onto an anisotropic metamaterial with an incident angle of θi, two opposite spin components of the transmitted beam will be separated in z = 0 plane due to the spatial spin splitting ΔY ± . Owing to the angular spin splitting, the spin separation Δ ± will change with the propagation distance. The inset shows the structure of the metamaterial.

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3. The spatial and angular spin splitting

The phase parameter φ plays an important role in the spin splitting. The spatial and angular spin splitting are proportional to cosφ and sinφ, respectively. Figure 2(a) gives the phase φ as functions of incident angle θi and metamaterial thickness d, when the parallel and perpendicular components of permittivity of metamaterial ε// = 1, ε = 0.01, respectively. By adjusting d and θi, φ can take arbitrary value between –π and + π. The phase φ varies more rapidly with incident angle θi for a thicker (larger d) metamaterial. The p-plane wave will undergo frustrated total internal reflection when θi>5.74°. Near the critical angle 5.74°, the transmission coefficients tpθi changes rapidly, which results in the fast change of phase parameter φ.

 figure: Fig. 2

Fig. 2 The changes of phase parameter φ (a), normalized spatial spin splitting ΔY/w0 (b), and normalized angular spin splitting Δθ/θd (c) as functions of incident angle θi and thickness of metamaterial d. In calculation, ε// = 1, ε = 0.01, w0 = 40λ.

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Although the phase parameter φ is independent of incident beam waist w0, the spatial and angular spin splitting ΔY and Δθ have strong dependences on it. When w0 = 40λ, λ being the free space wavelength, the change of normalized spatial (ΔY/w0) and normalized angular (Δθ/θd) spin splitting with θi and d are shown in Figs. 2(b) and 2(c), respectively. One can see from Figs. 2(b) and 2(c) that, both ΔY and Δθ are very small in the in upper-right corner offigures. This is because in this area, |tpθi| almost vanishes, thus is negligible comparing with |M|. Therefore, both ΔY and Δθ are near zero, which is directly seen from Eq. (5) and (6). When the metamaterial thickness or incident angle is small, ΔY and Δθ are both tiny small, because |tpθi| is equal to 1, which is much larger than |M|. When θi is around 7.1°, ΔY and Δθ can be quite large. As the phase φ increases with thickness d, see Fig. 2(a), both ΔY and Δθ change sign alternately. Around d = 0.1λ, the transmitted beam undergoes positive spatial spin splitting but negative angular spin splitting. With the increase of incident angle, φ trends to –π/2, therefore, ΔY trends to zero, and Δθ will trend close to θd. When d = 0.08λ, θi = 67.2°, Δθ = −0.99θd, which is the maximum angular spin splitting. The maximum spatial spin splitting is ΔY = 0.99w0, obtained at d = 0.42λ, θi = 11.4°, where the phase φ = −1° and |tpθi/M|=0.96. Near the maximum spatial splitting, there is a peak of angular spin splitting, where Δθ = 0.96θd, d = 0.78λ, θi = 7.5°, φ = 89°, and |tpθi/M|=1. Therefore, one concludes that, by adjusting d and θi, the spatial and angular spin splitting can approach closely to their upper limits.

The incident beam waist w0 will affect the spatial and angular spin splitting. The dependences of normalized spatial spin splitting ΔY/w0 with incident angle θi for w0 = 20λ, 40λ, 100λ, 200λ, 500λ are shown respectively in Fig. 3(a), where the thickness of metamaterial d is fixed to 0.42λ, which is the optimal thickness for w0 = 40λ. For each w0, there is a maximum splitting peak. The positions (incident angles θi) of the peaks increase monotonically with w0 owing to the change of the condition (1) for maximum spin splitting. The maximum spatial spin splitting ΔY are larger than 0.99w0 for the cases of w0 = 40λ and 100λ. It decreases slightly to 0.98w0 for w0 = 200λ. When w0 = 20λ and 500λ, ΔY = 0.95w0 and 0.96 w0, respectively. For the normalized angular spin splitting Δθ/θd, the positions of maximum peaks also increase with beam waist w0, as shown in Fig. 3(b), where d is fixed to 0.78λ. The maximum angular spin splitting Δθ for w0 = 40λ and 100λ are equal to 0.96θd and 0.97θd, respectively. However, it is smaller than 0.9θd when w0 = 20λ and 500λ. This is partly because the phase φ changes rapidly with incident angle θi when d = 0.78λ. It is worth to point out that, the spatial and angular spin splitting for each w0 should be optimized independently by adjusting d and θi, they both can almost reach their upper limits for w0>20λ.

 figure: Fig. 3

Fig. 3 The dependences of normalized spatial spin splitting ΔY/w0 (a), and normalized angular spin splitting Δθ/θd (b) on incident angle θi for w0 = 20λ (blue color), 40λ (orange color), 100λ (yellow color), 200λ (purple color), 500λ (light green color), respectively.

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Most metamaterials are absorptive medium in practical [20]. Here, we set the perpendicular component of permittivity ε as a complex [23]. The imagine part of ε indicates loss. A slightly loss Im[ε]<0.001 has little influence on both the normalized spatial (ΔY/w0) and normalized angular (Δθ/θd) spin splitting. For a larger loss with Im[ε] = 0.003, ΔY/w0 and Δθ/θd decrease, as shown in Figs. 4(a) and 4(b). When the real and image parts of ε are equal, Re[ε] = Im[ε] = 0.01, the image part of epsilon will give strong influence to the transmission coefficient tpθi, and the spatial and angular spin splitting become negative. This can be understood as following. As shown by Eqs. (5) and (6), the spatial and angular spin splitting ΔY and Δθ change with the phase parameter φ according to cosine and sine functions, respectively. The phase parameter φ is determined by the Fresnel transmission coefficients tpθi and tsθi. Specially, φ is approximately equal to the phase difference between tsθi and tpθi when |tpθi/tsθi|<<1, which holds for incident angle θi>5.74°. Since the phase difference can be modulated by the material loss Im[ε], the phase φ will vary with Im[ε]. Therefore, when Im[ε] increases from 0 to 0.01, both the spatial and angular spin splitting change signs [24], as shown by Figs. 4(a) and 4(b). When Im[ε] increases further, the absorption becomes the dominated effect. The amplitude and phase of tpθi vary slowly with incident angle. Therefore, ΔY/w0 and Δθ/θd, become much smoother. One finds from Figs. 4(a) and 4(b) that, with the increase of Im[ε], the positions of the peaks of both spatial and angular spin splitting increase gradually, which is caused by the increase of |tpθi|.

 figure: Fig. 4

Fig. 4 The dependences of normalized spatial spin splitting ΔY/w0 (a), and normalized angular spin splitting Δθ/θd (b) on incident angle θi forIm[ε] = 0 (blue color), 0.003 (orange color), 0.01 (yellow color), 0.03 (purple color), 0.1 (light green color), respectively. In calculation, Re[ε] = 0.01.

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We now investigate the changes of normalized spatial and angular spin splitting (ΔY/w0 and Δθ/θd) with real part of the perpendicular component of permittivity Re[ε]. As shown in Figs. 5(a) and 5(b), when Re[ε] increases from 0.001 to 0.1, both ΔY/w0 and Δθ/θd have similar profiles excepting for a relatively movement and an expansion. Interestingly, Δθ/θd for Re[ε] = 0.05 and 0.1 reach 0.99, which is larger than that for Re[ε] = 0.01 even though the thickness of metamaterial is optimized for Re[ε] = 0.01. By comparing Figs. 4 and 5, one can conclude that the image part of the perpendicular component of permittivity has stronger influences on the normalized spatial and angular spin splitting than the real part.

 figure: Fig. 5

Fig. 5 The same as Fig. 4 excepting for Re[ε] = 0 (blue color), 0.005 (orange color), 0.01 (yellow color), 0.05 (purple color), 0.1 (light green color), respectively. In calculation, Im[ε] = 0.

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4. The spin splitting of propagated beam

When the transmitted Gaussian beam through an anisotropic metamaterial propagates forward, the separation of two opposite spin components (Δz = Δ+-) will changes with the propagation distance z. The spin splitting of propagated beam Δz is sum of spatial spin splitting (ΔY) and angular-related spin splitting (zΔθ). For z<<z0, the angular-related spin splitting can be neglected, and the spin separation is mainly determined by spatial spin splitting. When z>>z0, however, ΔY is negligible, the spin separation Δz is equal to zΔθ, which increases linearly with propagation distance z. The size of transmitted beam will also increase with z. For a standard Gaussian beam, the radius of spot size is w(z) = w0[1 + (z/z0)2]1/2. Therefore, the spin splitting should be compared with w(z).

The changes of spin dependent displacements Δ ± with propagation distance z for three different cases are shown in Fig. 6(a). The first two cases correspond respectively to the maximum spatial and angular spin splitting for w0 = 40λ, which have been obtained in section 3. In the case 1, ΔY = 0.99w0 = 39.6λ, and Δθ = −0.0174θd = 1.38 × 10−4rad. Therefore, two opposite spin components of transmitted beam in z = 0 plane are nearly separated, which is clear in Fig. 6(b). Since the spatial and angular spin splitting are of opposite signs, the spin splitting Δz will decrease very slowly with z. In the case 2, ΔY = 0.014w0 = 0.56λ, and Δθ = −0.96θd = 7.64 × 10−3rad. The angular spin splitting is very close to its upper limit. Although the spin splitting in z = 0 plane is very small, but it increases rapidly with z. For z>3z0, it approaches the spot size of Gaussian beam w(z). So, two opposite spin components of transmitted beam are nearly resolvable after 3z0. However, the spin separations in cases 1 and 2 are smaller than the Gaussian beam size in the region of 0<z<3z0. In this region, we can maximize Δz by optimizing incident angle θi and thickness of metamaterial d. The upper limit of Δz is found to be the Gaussian beam size w(z). In case 3, we maximize Δz for z = z0, we get Δz = 0.99w(z0) = 56λ, with d = 0.61λ, θi = 8.66°. The spatial and angular spin splitting are ΔY = 0.70w0, and Δθ = 0.69θd. Although the spin splitting Δz in this case is smaller than that of case 1 at the beginning, it grows faster, and overtakes at z = 0.43z0. Comparing with the case 2, the spin splitting of case 3 is smaller after z = 2.5z0, However, in the vicinity of z = z0, the spin splitting in case 3 are the largest one. At z = z0 plane, the spin splitting of case 1 and 2 are identical. While at z = 2.5z0 plane, the spin splitting of case 2 and 3 are the same.

 figure: Fig. 6

Fig. 6 (a) The dependences of spin dependent displacements Δ+ (solid line) and Δ- (dashed line) on the propagation distance z for three different cases. In case 1 (red color), d = 0.42λ, θi = 11.4°. In case 2 (blue color), d = 0.78λ, θi = 7.5°. In case 3 (green color), d = 0.61λ, θi = 8.7°. The black lines correspond to the upper limit of spin splitting of propagated beam, and the distance between two black dashed lines is equal to the radius of spot size of Gaussian beam size w(z). (b) The intensity distributions of RCP and LCP components of transmitted beam in z = 0, z0, and 2.5z0 planes for three different cases.

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The spin splitting of transmitted beam is evident in Fig. 6(b), where the intensity distributions of RCP and LCP components in z = 0, z0, and 2.5z0 planes are shown for three different cases. The intensity profiles of RCP components shift toward + y direction, while those of LCP toward –y direction. The RCP and LCP filed components are well separated in the case 1 for z = 0 plane and in the case 3 for z = z0 plane, where the spin splitting Δz are equal to the spot size of Gaussian beam w(z). It is worth to notice that, the transmitted beam undergoes longitudinal displacements (along x-axis) in addition to the transverse one (along y-axis). The longitudinal displacements are caused by Goos-Hänchen shift, thus are spin independent. In all the cases in Fig. 6(b), the longitudinal displacements are negative, which results in the tilts of intensity profiles of both RCP and LCP field components.

5. Conclusions

In conclusion, we have found that the upper limits of spatial and angular spin splitting (ΔY and Δθ) of transmitted beam through an anisotropic metamaterial are equal to the beam waist w0 and divergent angle θd of the incident Gaussian beam, respectively. The influences of beam waist and the permittivity of metamaterial on ΔY and Δθ for fixed thickness d have been demonstrated. The spin splitting of propagated beam Δz is the combination of spatial and angular spin splitting. By optimizing d and θi, Δz for a given propagation distance z can be up to the spot size of Gaussian beam w(z). We believe these results are useful in developing spin-based nanophotonic applications [25,26].

Funding

National Natural Science Foundation of China (61675092); Natural Science Foundation of Guangdong Province (2016A030313).

References and links

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

2. X. J. Tan and X. S. Zhu, “Enhancing photonic spin Hall effect via long-range surface plasmon resonance,” Opt. Lett. 41(11), 2478–2481 (2016). [CrossRef]   [PubMed]  

3. X. Qiu, L. Xie, J. Qiu, Z. Zhang, J. Du, and F. Gao, “Diffraction-dependent spin splitting in spin Hall effect of light on reflection,” Opt. Express 23(15), 18823–18831 (2015). [CrossRef]   [PubMed]  

4. X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015). [CrossRef]   [PubMed]  

5. 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]  

6. 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]  

7. 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]  

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. N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36(16), 3200–3202 (2011). [CrossRef]   [PubMed]  

10. 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]  

11. X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015). [CrossRef]   [PubMed]  

12. J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014). [CrossRef]   [PubMed]  

13. J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015). [CrossRef]  

14. H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010). [CrossRef]  

15. X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014). [CrossRef]   [PubMed]  

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

17. 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]  

18. S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Lett. 39(21), 6229–6232 (2014). [CrossRef]   [PubMed]  

19. G. Jayaswal, G. Mistura, and M. Merano, “Observing angular deviations in light-beam reflection via weak measurements,” Opt. Lett. 39(21), 6257–6260 (2014). [CrossRef]   [PubMed]  

20. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013). [CrossRef]  

21. P. Ginzburg, F. J. R. Fortuño, G. A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R. J. Pollard, I. Iorsh, A. Atrashchenko, P. A. Belov, Y. S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, and A. V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21(12), 14907–14917 (2013). [CrossRef]   [PubMed]  

22. O. J. S. Santana, S. A. Carvalho, S. De Leo, and L. E. de Araujo, “Weak measurement of the composite Goos-Hänchen shift in the critical region,” Opt. Lett. 41(16), 3884–3887 (2016). [CrossRef]   [PubMed]  

23. Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015). [CrossRef]   [PubMed]  

24. H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011). [CrossRef]  

25. A. Shaltout, J. Liu, A. Kildishev, and V. Shalaev, “Photonic spin Hall effect in gap–plasmon metasurfaces for on-chip chiroptical spectroscopy,” Optica 2(10), 860 (2015). [CrossRef]  

26. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]  

References

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  1. K. Y. Bliokh and A. Aiello, “Goos-hächen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
    [Crossref]
  2. X. J. Tan and X. S. Zhu, “Enhancing photonic spin Hall effect via long-range surface plasmon resonance,” Opt. Lett. 41(11), 2478–2481 (2016).
    [Crossref] [PubMed]
  3. X. Qiu, L. Xie, J. Qiu, Z. Zhang, J. Du, and F. Gao, “Diffraction-dependent spin splitting in spin Hall effect of light on reflection,” Opt. Express 23(15), 18823–18831 (2015).
    [Crossref] [PubMed]
  4. X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015).
    [Crossref] [PubMed]
  5. 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]
  6. 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]
  7. 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]
  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. N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36(16), 3200–3202 (2011).
    [Crossref] [PubMed]
  10. 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]
  11. X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015).
    [Crossref] [PubMed]
  12. J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014).
    [Crossref] [PubMed]
  13. J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
    [Crossref]
  14. H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
    [Crossref]
  15. X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
    [Crossref] [PubMed]
  16. T. Tang, C. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6, 30762 (2016).
    [Crossref] [PubMed]
  17. 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]
  18. S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Lett. 39(21), 6229–6232 (2014).
    [Crossref] [PubMed]
  19. G. Jayaswal, G. Mistura, and M. Merano, “Observing angular deviations in light-beam reflection via weak measurements,” Opt. Lett. 39(21), 6257–6260 (2014).
    [Crossref] [PubMed]
  20. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013).
    [Crossref]
  21. P. Ginzburg, F. J. R. Fortuño, G. A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R. J. Pollard, I. Iorsh, A. Atrashchenko, P. A. Belov, Y. S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, and A. V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21(12), 14907–14917 (2013).
    [Crossref] [PubMed]
  22. O. J. S. Santana, S. A. Carvalho, S. De Leo, and L. E. de Araujo, “Weak measurement of the composite Goos-Hänchen shift in the critical region,” Opt. Lett. 41(16), 3884–3887 (2016).
    [Crossref] [PubMed]
  23. Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
    [Crossref] [PubMed]
  24. H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
    [Crossref]
  25. A. Shaltout, J. Liu, A. Kildishev, and V. Shalaev, “Photonic spin Hall effect in gap–plasmon metasurfaces for on-chip chiroptical spectroscopy,” Optica 2(10), 860 (2015).
    [Crossref]
  26. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
    [Crossref]

2016 (4)

2015 (9)

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

A. Shaltout, J. Liu, A. Kildishev, and V. Shalaev, “Photonic spin Hall effect in gap–plasmon metasurfaces for on-chip chiroptical spectroscopy,” Optica 2(10), 860 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

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]

X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015).
[Crossref] [PubMed]

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

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]

X. Qiu, L. Xie, J. Qiu, Z. Zhang, J. Du, and F. Gao, “Diffraction-dependent spin splitting in spin Hall effect of light on reflection,” Opt. Express 23(15), 18823–18831 (2015).
[Crossref] [PubMed]

X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015).
[Crossref] [PubMed]

2014 (4)

J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014).
[Crossref] [PubMed]

S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Lett. 39(21), 6229–6232 (2014).
[Crossref] [PubMed]

G. Jayaswal, G. Mistura, and M. Merano, “Observing angular deviations in light-beam reflection via weak measurements,” Opt. Lett. 39(21), 6257–6260 (2014).
[Crossref] [PubMed]

X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
[Crossref] [PubMed]

2013 (3)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013).
[Crossref]

P. Ginzburg, F. J. R. Fortuño, G. A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R. J. Pollard, I. Iorsh, A. Atrashchenko, P. A. Belov, Y. S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, and A. V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21(12), 14907–14917 (2013).
[Crossref] [PubMed]

K. Y. Bliokh and A. Aiello, “Goos-hächen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

2011 (3)

N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36(16), 3200–3202 (2011).
[Crossref] [PubMed]

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]

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

2010 (1)

H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
[Crossref]

2008 (1)

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

2006 (1)

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]

Aiello, A.

K. Y. Bliokh and A. Aiello, “Goos-hächen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36(16), 3200–3202 (2011).
[Crossref] [PubMed]

Ankonina, G.

Atrashchenko, A.

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013).
[Crossref]

Belov, P. A.

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-hächen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

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]

Bliokh, Y. P.

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]

Carvalho, S. A.

Chan, C. T.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Chen, H.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

de Araujo, L. E.

De Leo, S.

Dennis, M. R.

J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014).
[Crossref] [PubMed]

Dickson, W.

Du, J.

Fan, D.

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]

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
[Crossref]

Fortuño, F. J. R.

Gao, F.

Ghosh, N.

Ginzburg, P.

Gong, Q.

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

Goswami, S.

Götte, J. B.

J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014).
[Crossref] [PubMed]

Grosche, S.

Hermosa, N.

Hosten, O.

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

Iorsh, I.

Jayaswal, G.

Kildishev, A.

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013).
[Crossref]

Kivshar, Y. S.

Kwiat, P.

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

Li, C.

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

Li, Y.

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

Ling, X.

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]

X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
[Crossref] [PubMed]

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

Liu, J.

Löffler, W.

J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014).
[Crossref] [PubMed]

Luo, H.

X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
[Crossref] [PubMed]

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]

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
[Crossref]

Luo, L.

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

Merano, M.

Mistura, G.

Morgan, F.

Murphy, A.

Nandi, A.

Nevet, A.

Nori, F.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Nugrowati, A. M.

Orenstein, M.

Ornigotti, M.

Pal, M.

Panigrahi, P. K.

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013).
[Crossref]

Pollard, R. J.

Qiu, J.

Qiu, X.

Ren, J. L.

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

Rodríguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Santana, O. J. S.

Shalaev, V.

Shaltout, A.

She, W.

Shu, W.

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

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]

H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
[Crossref]

Szameit, A.

Tan, X. J.

Tang, T.

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

Wang, B.

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

Wen, S.

X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
[Crossref] [PubMed]

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]

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
[Crossref]

Woerdman, J. P.

Wurtz, G. A.

Xiao, Y. F.

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

Xie, L.

Xu, Y.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Zayats, A. V.

Zhang, Z.

Zhou, X.

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]

X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
[Crossref] [PubMed]

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]

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

Zhu, W.

Zhu, X. S.

Appl. Phys. Lett. (1)

J. L. Ren, B. Wang, Y. F. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 54–59 (2015).
[Crossref]

J. Opt. (1)

K. Y. Bliokh and A. Aiello, “Goos-hächen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

Nat. Photonics (2)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Corrigendum: Hyperbolic metamaterials,” Nat. Photonics 8(1), 78 (2013).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Opt. Express (4)

Opt. Lett. (8)

X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015).
[Crossref] [PubMed]

X. J. Tan and X. S. Zhu, “Enhancing photonic spin Hall effect via long-range surface plasmon resonance,” Opt. Lett. 41(11), 2478–2481 (2016).
[Crossref] [PubMed]

N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36(16), 3200–3202 (2011).
[Crossref] [PubMed]

X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015).
[Crossref] [PubMed]

O. J. S. Santana, S. A. Carvalho, S. De Leo, and L. E. de Araujo, “Weak measurement of the composite Goos-Hänchen shift in the critical region,” Opt. Lett. 41(16), 3884–3887 (2016).
[Crossref] [PubMed]

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]

S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Lett. 39(21), 6229–6232 (2014).
[Crossref] [PubMed]

G. Jayaswal, G. Mistura, and M. Merano, “Observing angular deviations in light-beam reflection via weak measurements,” Opt. Lett. 39(21), 6257–6260 (2014).
[Crossref] [PubMed]

Optica (1)

Phys. Rev. A (3)

H. Luo, X. Ling, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhancing or suppressing the spin Hall effect of light in layered nanostructures,” Phys. Rev. A 84(3), 033801 (2011).
[Crossref]

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]

H. Luo, S. Wen, W. Shu, and D. Fan, “Spin Hall effect of light in photon tunneling,” Phys. Rev. A 82(4), 043825 (2010).
[Crossref]

Phys. Rev. Lett. (2)

J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112(23), 233901 (2014).
[Crossref] [PubMed]

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]

Sci. Rep. (3)

X. Zhou, X. Ling, Z. Zhang, H. Luo, and S. Wen, “Observation of spin Hall effect in photon tunneling via weak measurements,” Sci. Rep. 4, 7388 (2014).
[Crossref] [PubMed]

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

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Science (1)

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

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Figures (6)

Fig. 1
Fig. 1 The schematic of spin splitting of transmitted beam. A linearly polarized Gaussian beam is incident onto an anisotropic metamaterial with an incident angle of θi, two opposite spin components of the transmitted beam will be separated in z = 0 plane due to the spatial spin splitting ΔY ± . Owing to the angular spin splitting, the spin separation Δ ± will change with the propagation distance. The inset shows the structure of the metamaterial.
Fig. 2
Fig. 2 The changes of phase parameter φ (a), normalized spatial spin splitting ΔY/w0 (b), and normalized angular spin splitting Δθ/θd (c) as functions of incident angle θi and thickness of metamaterial d. In calculation, ε// = 1, ε = 0.01, w0 = 40λ.
Fig. 3
Fig. 3 The dependences of normalized spatial spin splitting ΔY/w0 (a), and normalized angular spin splitting Δθ/θd (b) on incident angle θi for w0 = 20λ (blue color), 40λ (orange color), 100λ (yellow color), 200λ (purple color), 500λ (light green color), respectively.
Fig. 4
Fig. 4 The dependences of normalized spatial spin splitting ΔY/w0 (a), and normalized angular spin splitting Δθ/θd (b) on incident angle θi for Im [ ε ] = 0 (blue color), 0.003 (orange color), 0.01 (yellow color), 0.03 (purple color), 0.1 (light green color), respectively. In calculation, Re [ ε ] = 0.01.
Fig. 5
Fig. 5 The same as Fig. 4 excepting for Re [ ε ] = 0 (blue color), 0.005 (orange color), 0.01 (yellow color), 0.05 (purple color), 0.1 (light green color), respectively. In calculation, Im [ ε ] = 0.
Fig. 6
Fig. 6 (a) The dependences of spin dependent displacements Δ+ (solid line) and Δ- (dashed line) on the propagation distance z for three different cases. In case 1 (red color), d = 0.42λ, θi = 11.4°. In case 2 (blue color), d = 0.78λ, θi = 7.5°. In case 3 (green color), d = 0.61λ, θi = 8.7°. The black lines correspond to the upper limit of spin splitting of propagated beam, and the distance between two black dashed lines is equal to the radius of spot size of Gaussian beam size w(z). (b) The intensity distributions of RCP and LCP components of transmitted beam in z = 0, z0, and 2.5z0 planes for three different cases.

Equations (9)

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E ˜ t = w 0 2 π exp [ ( k x 2 + k y 2 ) w 0 2 4 ] { [ t p θ i + k x k 0 t p θ i ] e ^ x + [ k y k 0 ( t p θ i t s θ i ) cot θ i ] e ^ y } ,
t p θ i = 1 cos δ p i sin δ p 2 ε ( 1 + cos 2 θ i ) sin 2 θ i cos θ i ε / / ε ( ε sin 2 θ i ) ,
t s θ i = 1 cos δ s i sin δ s 2 cos 2 θ i + ε / / cos θ i ε / / sin 2 θ i ,
E ± = 1 π w 0 z 0 z 0 + i z exp [ k 0 ( x 2 + y 2 ) 2 ( z 0 + i z ) ] [ t p θ i + i x z 0 + i z t p θ i ± y ( t p θ i t s θ i ) cot θ i z 0 + i z ] e ^ ± ,
Δ ± = y | E ± | 2 d x d y / | E ± | 2 d x d y .
Δ Y ± = ± w 0 | t p θ i | | M | cos φ | t p θ i | 2 + 1 k 0 2 w 0 2 | t p θ i | 2 + | M | 2 ,
Δ θ ± = ± 2 k 0 w 0 | t p θ i | | M | sin φ | t p θ i | 2 + 1 k 0 2 w 0 2 | t p θ i | 2 + | M | 2 ,
| Δ Y | m = w 0 1 + 1 2 | t p θ i | 2 k 0 2 w 0 2 | t p θ i | 2 .
| Δ θ | m = 2 k 0 w 0 1 1 + 1 2 | t p θ i | 2 k 0 2 w 0 2 | t p θ i | 2 .

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