Large spatial and angular spin splitting in a thin anisotropic ε-near-zero metamaterial

Wenguo Zhu; Jianhui Yu; Heyuan Guan; Huihui Lu; Jieyuan Tang; Yuanhan Luo; Zhe Chen

doi:10.1364/OE.25.005196

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 θ_i and thickness of metamaterial d, which results in the spatial and angular spin splitting. By controlling θ_i 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 θ_i. 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 ${\tilde{E}}_{i} = w_{0} / {(2 π)}^{1 / 2} \exp [- (k_{x}^{2} + k_{y}^{2}) w_{0}^{2} / 4] {\hat{e}}_{x}$ . According to Ref [1,17], the transmitted angular spectrum is given by

{\tilde{E}}_{t} = \frac{w_{0}}{\sqrt{2 π}} \exp [\frac{- (k_{x}^{2} + k_{y}^{2}) w_{0}^{2}}{4}] {[t_{p θ_{i}} + \frac{k_{x}}{k_{0}} \frac{\partial t_{p}}{\partial θ_{i}}] {\hat{e}}_{x} + [\frac{k_{y}}{k_{0}} (t_{p θ_{i}} - t_{s θ_{i}}) \cot θ_{i}] {\hat{e}}_{y}},

where k₀ is wavenumber in free space;

t_{p θ_{i}}

and

t_{s θ_{i}}

are the Fresnel transmission coefficients for p- and s- waves incident at θ_i; ∂t_p/θ_i is the first-order derivative of transmission coefficient. For a thin anisotropic metamaterial, the Fresnel transmission coefficients are in form of

t_{p θ_{i}} = \frac{1}{\cos δ_{p} - \frac{i \sin δ_{p}}{2} \frac{ε_{⊥} (1 + \cos^{2} θ_{i}) - \sin^{2} θ_{i}}{\cos θ_{i} \sqrt{ε_{/ /} ε_{⊥} (ε_{⊥} - \sin^{2} θ_{i})}}},

t_{s θ_{i}} = \frac{1}{\cos δ_{s} - \frac{i \sin δ_{s}}{2} \frac{\cos 2 θ_{i} + ε_{/ /}}{\cos θ_{i} \sqrt{ε_{/ /} - \sin^{2} θ_{i}}}},

where δ_p = k₀d(ε_//-ε_//sin²θ_i/

ε_{⊥}

)^1/2, δ_s = k₀d(ε_//-sin²θ_i)^1/2 with d being the thickness of metamaterial. When ε_// = 1,

t_{s θ_{i}}

= exp[ik₀d(1-sin²θ_i)^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 _± = ê_x ± iê_y, the transmitted field is

E_{\pm} = \frac{1}{\sqrt{π} w_{0}} \frac{z_{0}}{z_{0} + i z} \exp [- \frac{k_{0} (x^{2} + y^{2})}{2 (z_{0} + i z)}] [t_{p θ_{i}} + \frac{i x}{z_{0} + i z} \frac{\partial t_{p}}{\partial θ_{i}} \pm \frac{y (t_{p θ_{i}} - t_{s θ_{i}}) \cot θ_{i}}{z_{0} + i z}] {\hat{e}}_{\pm},

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_{\pm} = \iint y | E_{\pm} |^{2} d x d y / \iint | E_{\pm} |^{2} d x d y .

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

Δ Y_{\pm} = \pm w_{0} \frac{| t_{p θ_{i}} | | M | \cos φ}{| t_{p θ_{i}} |^{2} + \frac{1}{k_{0}^{2} w_{0}^{2}} | \frac{\partial t_{p}}{\partial θ_{i}} |^{2} + | M |^{2}},

Δ θ_{\pm} = \pm \frac{2}{k_{0} w_{0}} \frac{| t_{p θ_{i}} | | M | \sin φ}{| t_{p θ_{i}} |^{2} + \frac{1}{k_{0}^{2} w_{0}^{2}} | \frac{\partial t_{p}}{\partial θ_{i}} |^{2} + | M |^{2}},

where

M = (t_{p θ_{i}} - t_{s θ_{i}}) \cot θ_{i} / k_{0} w_{0}

,

φ = \arg [t_{p θ_{i}}^{*} 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₊-ΔY_- and Δθ = Δθ₊-Δθ_-, 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,

| t_{p θ_{i}} | > > | \partial t_{p} / \partial θ_{i} | / k_{0}^{} w_{0}^{}

, so the term

| \partial t_{p} / \partial θ_{i} |^{2} / k_{0}^{2} w_{0}^{2}

in the denominator of Eq. (5) can be neglected. Therefore, the maximum spatial spin splitting is approximately obtained when the conditions: 1)

| t_{p θ_{i}} |^{2} = | M |^{2}

, and 2) φ = 0 or π are satisfied simultaneously. These two conditions ensure

| t_{p θ_{i}} | | M | / [| t_{p θ_{i}} |^{2} + | M |^{2}]

and |cosφ| reaching their maximum values, respectively. Under these conditions, the spatial spin splitting of transmitted beam is given by

| Δ Y |_{m} = \frac{w_{0}}{1 + \frac{1}{2 | t_{p θ_{i}} |^{2} k_{0}^{2} w_{0}^{2}} | \frac{\partial t_{p}}{\partial θ_{i}} |^{2}} .

Since

| t_{p θ_{i}} | > > | \partial t_{p} / \partial θ_{i} | / k_{0}^{} w_{0}^{}

, |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

| Δ θ |_{m} = \frac{2}{k_{0} w_{0}} \frac{1}{1 + \frac{1}{2 | t_{p θ_{i}} |^{2} k_{0}^{2} w_{0}^{2}} | \frac{\partial t_{p}}{\partial θ_{i}} |^{2}} .

Similarly, the upper limit of spin angular splitting is the divergent angle of incident Gaussian beam θ_d = 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.

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 θ_i, 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 θ_i and metamaterial thickness d, when the parallel and perpendicular components of permittivity of metamaterial ε_// = 1, $ε_{⊥}$ = 0.01, respectively. By adjusting d and θ_i, φ can take arbitrary value between –π and + π. The phase φ varies more rapidly with incident angle θ_i for a thicker (larger d) metamaterial. The p-plane wave will undergo frustrated total internal reflection when θ_i>5.74°. Near the critical angle 5.74°, the transmission coefficients $t_{p θ_{i}}$ changes rapidly, which results in the fast change of phase parameter φ.

Fig. 2 The changes of phase parameter φ (a), normalized spatial spin splitting ΔY/w₀ (b), and normalized angular spin splitting Δθ/θ_d (c) as functions of incident angle θ_i 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/θ_d) spin splitting with θ_i 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, $| t_{p θ_{i}} |$ 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 $| t_{p θ_{i}} |$ is equal to 1, which is much larger than |M|. When θ_i 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 θ_d. When d = 0.08λ, θ_i = 67.2°, Δθ = −0.99θ_d, which is the maximum angular spin splitting. The maximum spatial spin splitting is ΔY = 0.99w₀, obtained at d = 0.42λ, θ_i = 11.4°, where the phase φ = −1° and $| t_{p θ_{i}} / M | = 0.96$ . Near the maximum spatial splitting, there is a peak of angular spin splitting, where Δθ = 0.96θ_d, d = 0.78λ, θ_i = 7.5°, φ = 89°, and $| t_{p θ_{i}} / M | = 1$ . Therefore, one concludes that, by adjusting d and θ_i, 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 θ_i 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 θ_i) 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/θ_d, 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 0.96θ_d and 0.97θ_d, respectively. However, it is smaller than 0.9θ_d when w₀ = 20λ and 500λ. This is partly because the phase φ changes rapidly with incident angle θ_i 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 θ_i, they both can almost reach their upper limits for w₀>20λ.

Fig. 3 The dependences of normalized spatial spin splitting ΔY/w₀ (a), and normalized angular spin splitting Δθ/θ_d (b) on incident angle θ_i 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/θ_d) spin splitting. For a larger loss with $Im [ε_{⊥}]$ = 0.003, ΔY/w₀ and Δθ/θ_d 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 $t_{p θ_{i}}$ , 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 $t_{p θ_{i}}$ and $t_{s θ_{i}}$ . Specially, φ is approximately equal to the phase difference between $t_{s θ_{i}}$ and $t_{p θ_{i}}$ when $| t_{p θ_{i}} / t_{s θ_{i}} | < < 1$ , which holds for incident angle θ_i>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 $t_{p θ_{i}}$ vary slowly with incident angle. Therefore, ΔY/w₀ and Δθ/θ_d, 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 $| t_{p θ_{i}} |$ .

Fig. 4 The dependences of normalized spatial spin splitting ΔY/w₀ (a), and normalized angular spin splitting Δθ/θ_d (b) on incident angle θ_i 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.

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We now investigate the changes of normalized spatial and angular spin splitting (ΔY/w₀ and Δθ/θ_d) 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/θ_d have similar profiles excepting for a relatively movement and an expansion. Interestingly, Δθ/θ_d 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.

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 (Δ_z = Δ₊-Δ_-) will changes with the propagation distance z. The spin splitting of propagated beam Δ_z 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 Δ_z 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 = −0.0174θ_d = 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 Δ_z will decrease very slowly with z. In the case 2, ΔY = 0.014w₀ = 0.56λ, and Δθ = −0.96θ_d = 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 Δ_z by optimizing incident angle θ_i and thickness of metamaterial d. The upper limit of Δ_z is found to be the Gaussian beam size w(z). In case 3, we maximize Δ_z for z = z₀, we get Δ_z = 0.99w(z₀) = 56λ, with d = 0.61λ, θ_i = 8.66°. The spatial and angular spin splitting are ΔY = 0.70w₀, and Δθ = 0.69θ_d. Although the spin splitting Δ_z 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.

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, θ_i = 11.4°. In case 2 (blue color), d = 0.78λ, θ_i = 7.5°. In case 3 (green color), d = 0.61λ, θ_i = 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 Δ_z 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 θ_d 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 Δ_z is the combination of spatial and angular spin splitting. By optimizing d and θ_i, Δ_z 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).

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Large spatial and angular spin splitting in a thin anisotropic ε-near-zero metamaterial

Abstract

1. Introduction

2. Theory and model

3. The spatial and angular spin splitting

4. The spin splitting of propagated beam

5. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (9)

Optics Express