## 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 ${\epsilon}_{\perp}$ 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*and

_{i}*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*_{//}, ${\epsilon}_{\perp}$], where *ε*_{//} is determined by the permittivity of background dielectric, while ${\epsilon}_{\perp}$ 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 ${\epsilon}_{\perp}$ 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 ${\tilde{E}}_{i}={w}_{0}/{(2\pi )}^{1/2}\mathrm{exp}\left[-({k}_{x}^{2}+{k}_{y}^{2}){w}_{0}^{2}/4\right]{\widehat{e}}_{x}$. According to Ref [1,17], the transmitted angular spectrum is given by

*k*

_{0}is wavenumber in free space; ${t}_{p{\theta}_{i}}$ and ${t}_{s{\theta}_{i}}$ are the Fresnel transmission coefficients for

*p*- and

*s*- waves incident at

*θ*; ∂

_{i}*t*/

_{p}*θ*is the first-order derivative of transmission coefficient. For a thin anisotropic metamaterial, the Fresnel transmission coefficients are in form of

_{i}*δ*=

_{p}*k*

_{0}

*d*(

*ε*

_{//}-

*ε*

_{//}sin

^{2}

*θ*/${\epsilon}_{\perp}$)

_{i}^{1/2},

*δ*=

_{s}*k*

_{0}

*d*(

*ε*

_{//}-sin

^{2}

*θ*)

_{i}^{1/2}with

*d*being the thickness of metamaterial. When

*ε*

_{//}= 1, ${t}_{s{\theta}_{i}}$ = exp[

*ik*

_{0}

*d*(1-sin

^{2}

*θ*)

_{i}^{1/2}], as if the

*s*-plane wave is propagating in the free space. However, the transmission property of

*p*-wave varies with ${\epsilon}_{\perp}$. 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}*i*

**ê**

*, the transmitted field is*

_{y}**E**

_{±}are the right and left handed circular polarization (RCP and LCP) components;

*z*

_{0}=

*k*

_{0}

*w*

_{0}

^{2}/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]

*z*-dependent and

*z*-independent terms: Δ

_{±}= Δ

*Y*

_{±}+

*z*Δ

*θ*

_{±}[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

*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 Δ

*Y*= Δ

*Y*

_{+}-Δ

*Y*

_{-}and Δ

*θ*= Δ

*θ*

_{+}-Δ

*θ*

_{-}, respectively. Generally, the transmitted beam can have spatial and angular spin splitting simultaneously. When phase parameter

*φ*= ±

*π*/2, Δ

*Y*= 0, the transmitted beam only undergoes angular spin splitting. While

*φ*= 0 or

*π*, Δ

*θ*= 0, the transmitted beam splits spatially. In the paraxial region, $\left|{t}_{p{\theta}_{i}}\right|>>|\partial {t}_{p}/\partial {\theta}_{i}|/{k}_{0}^{}{w}_{0}^{}$, so the term $|\partial {t}_{p}/\partial {\theta}_{i}{|}^{2}/{k}_{0}^{2}{w}_{0}^{2}$ in the denominator of Eq. (5) can be neglected. Therefore, the maximum spatial spin splitting is approximately obtained when the conditions: 1) $|{t}_{p{\theta}_{i}}{|}^{2}=|M{|}^{2}$, and 2)

*φ*= 0 or

*π*are satisfied simultaneously. These two conditions ensure $|{t}_{p{\theta}_{i}}\left|\right|M|/[|{t}_{p{\theta}_{i}}{|}^{2}+|M{|}^{2}]$ and |cos

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

*Y*|

*approaches closely to*

_{m}*w*

_{0}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

*θ*= 2/

_{d}*k*

_{0}

*w*

_{0}. 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.

## 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 cos*φ* and sin*φ*, 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, ${\epsilon}_{\perp}$ = 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*for a thicker (larger

_{i}*d*) metamaterial. The

*p*-plane wave will undergo frustrated total internal reflection when

*θ*>5.74°. Near the critical angle 5.74°, the transmission coefficients ${t}_{p{\theta}_{i}}$ changes rapidly, which results in the fast change of phase parameter

_{i}*φ*.

Although the phase parameter *φ* is independent of incident beam waist *w*_{0}, the spatial and angular spin splitting Δ*Y* and Δ*θ* have strong dependences on it. When *w*_{0} = 40*λ*, *λ* being the free space wavelength, the change of normalized spatial (Δ*Y*/*w*_{0}) and normalized angular (Δ*θ*/*θ _{d}*) spin splitting with

*θ*and

_{i}*d*are shown in Figs. 2(b) and 2(c), respectively. One can see from Figs. 2(b) and 2(c) that, both Δ

*Y*and Δ

*θ*are very small in the in upper-right corner offigures. This is because in this area, $\left|{t}_{p{\theta}_{i}}\right|$ almost vanishes, thus is negligible comparing with |M|. Therefore, both Δ

*Y*and Δ

*θ*are near zero, which is directly seen from Eq. (5) and (6). When the metamaterial thickness or incident angle is small, Δ

*Y*and Δ

*θ*are both tiny small, because $\left|{t}_{p{\theta}_{i}}\right|$ is equal to 1, which is much larger than |M|. When

*θ*is around 7.1°, Δ

_{i}*Y*and Δ

*θ*can be quite large. As the phase

*φ*increases with thickness

*d*, see Fig. 2(a), both Δ

*Y*and Δ

*θ*change sign alternately. Around

*d*= 0.1

*λ*, 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, Δ

*Y*trends to zero, and Δ

*θ*will trend close to

*θ*. When

_{d}*d*= 0.08

*λ*,

*θ*= 67.2°, Δ

_{i}*θ*= −0.99

*θ*, which is the maximum angular spin splitting. The maximum spatial spin splitting is Δ

_{d}*Y*= 0.99

*w*

_{0}, obtained at

*d*= 0.42

*λ*,

*θ*= 11.4°, where the phase

_{i}*φ*= −1° and $|{t}_{p{\theta}_{i}}/M|=0.96$. Near the maximum spatial splitting, there is a peak of angular spin splitting, where Δ

*θ*= 0.96

*θ*, d = 0.78

_{d}*λ*,

*θ*= 7.5°,

_{i}*φ*= 89°, and $|{t}_{p{\theta}_{i}}/M|=1$. Therefore, one concludes that, by adjusting

*d*and

*θ*, the spatial and angular spin splitting can approach closely to their upper limits.

_{i}The incident beam waist *w*_{0} will affect the spatial and angular spin splitting. The dependences of normalized spatial spin splitting Δ*Y*/*w*_{0} with incident angle *θ _{i}* for

*w*

_{0}= 20

*λ*, 40

*λ*, 100

*λ*, 200

*λ*, 500

*λ*are shown respectively in Fig. 3(a), where the thickness of metamaterial

*d*is fixed to 0.42

*λ*, which is the optimal thickness for

*w*

_{0}= 40

*λ*. For each

*w*

_{0}, there is a maximum splitting peak. The positions (incident angles

*θ*) of the peaks increase monotonically with

_{i}*w*

_{0}owing to the change of the condition (1) for maximum spin splitting. The maximum spatial spin splitting Δ

*Y*are larger than 0.99

*w*

_{0}for the cases of

*w*

_{0}= 40

*λ*and 100

*λ*. It decreases slightly to 0.98

*w*

_{0}for

*w*

_{0}= 200

*λ*. When

*w*

_{0}= 20

*λ*and 500

*λ*, Δ

*Y*= 0.95

*w*

_{0}and 0.96

*w*

_{0}, respectively. For the normalized angular spin splitting Δ

*θ*/

*θ*, the positions of maximum peaks also increase with beam waist

_{d}*w*

_{0}, as shown in Fig. 3(b), where

*d*is fixed to 0.78

*λ*. The maximum angular spin splitting Δ

*θ*for

*w*

_{0}= 40

*λ*and 100

*λ*are equal to 0.96

*θ*and 0.97

_{d}*θ*, respectively. However, it is smaller than 0.9

_{d}*θ*when

_{d}*w*

_{0}= 20

*λ*and 500

*λ*. This is partly because the phase

*φ*changes rapidly with incident angle

*θ*when

_{i}*d*= 0.78

*λ*. It is worth to point out that, the spatial and angular spin splitting for each

*w*

_{0}should be optimized independently by adjusting

*d*and

*θ*, they both can almost reach their upper limits for

_{i}*w*

_{0}>20

*λ*.

Most metamaterials are absorptive medium in practical [20]. Here, we set the perpendicular component of permittivity ${\epsilon}_{\perp}$ as a complex [23]. The imagine part of ${\epsilon}_{\perp}$ indicates loss. A slightly loss $\mathrm{Im}[{\epsilon}_{\perp}]$<0.001 has little influence on both the normalized spatial (Δ*Y*/*w*_{0}) and normalized angular (Δ*θ*/*θ _{d}*) spin splitting. For a larger loss with $\mathrm{Im}[{\epsilon}_{\perp}]$ = 0.003, Δ

*Y*/

*w*

_{0}and Δ

*θ*/

*θ*decrease, as shown in Figs. 4(a) and 4(b). When the real and image parts of ${\epsilon}_{\perp}$ are equal, $\mathrm{Re}[{\epsilon}_{\perp}]$ = $\mathrm{Im}[{\epsilon}_{\perp}]$ = 0.01, the image part of epsilon will give strong influence to the transmission coefficient ${t}_{p{\theta}_{i}}$, 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 Δ

_{d}*Y*and Δ

*θ*change with the phase parameter

*φ*according to cosine and sine functions, respectively. The phase parameter

*φ*is determined by the Fresnel transmission coefficients ${t}_{p{\theta}_{i}}$ and ${t}_{s{\theta}_{i}}$. Specially,

*φ*is approximately equal to the phase difference between ${t}_{s{\theta}_{i}}$ and ${t}_{p{\theta}_{i}}$ when $|{t}_{p{\theta}_{i}}/{t}_{s{\theta}_{i}}|<<1$, which holds for incident angle

*θ*>5.74°. Since the phase difference can be modulated by the material loss $\mathrm{Im}[{\epsilon}_{\perp}]$, the phase

_{i}*φ*will vary with $\mathrm{Im}[{\epsilon}_{\perp}]$. Therefore, when $\mathrm{Im}[{\epsilon}_{\perp}]$ 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 $\mathrm{Im}[{\epsilon}_{\perp}]$ increases further, the absorption becomes the dominated effect. The amplitude and phase of ${t}_{p{\theta}_{i}}$ vary slowly with incident angle. Therefore, Δ

*Y*/

*w*

_{0}and Δ

*θ*/

*θ*, become much smoother. One finds from Figs. 4(a) and 4(b) that, with the increase of $\mathrm{Im}[{\epsilon}_{\perp}]$, the positions of the peaks of both spatial and angular spin splitting increase gradually, which is caused by the increase of $\left|{t}_{p{\theta}_{i}}\right|$.

_{d}We now investigate the changes of normalized spatial and angular spin splitting (Δ*Y*/*w*_{0} and Δ*θ*/*θ _{d}*) with real part of the perpendicular component of permittivity $\mathrm{Re}[{\epsilon}_{\perp}]$. As shown in Figs. 5(a) and 5(b), when $\mathrm{Re}[{\epsilon}_{\perp}]$ increases from 0.001 to 0.1, both Δ

*Y*/

*w*

_{0}and Δ

*θ*/

*θ*have similar profiles excepting for a relatively movement and an expansion. Interestingly, Δ

_{d}*θ*/

*θ*for $\mathrm{Re}[{\epsilon}_{\perp}]$ = 0.05 and 0.1 reach 0.99, which is larger than that for $\mathrm{Re}[{\epsilon}_{\perp}]$ = 0.01 even though the thickness of metamaterial is optimized for $\mathrm{Re}[{\epsilon}_{\perp}]$ = 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.

_{d}## 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

*is sum of spatial spin splitting (Δ*

_{z}*Y*) and angular-related spin splitting (

*z*Δ

*θ*). For

*z*<<

*z*

_{0}, the angular-related spin splitting can be neglected, and the spin separation is mainly determined by spatial spin splitting. When

*z*>>

*z*

_{0}, however, Δ

*Y*is negligible, the spin separation Δ

_{z}is equal to

*z*Δ

*θ*, 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*) =

*w*

_{0}[1 + (

*z*/

*z*

_{0})

^{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 *w*_{0} = 40*λ*, which have been obtained in section 3. In the case 1, Δ*Y* = 0.99*w*_{0} = 39.6*λ*, and Δ*θ* = −0.0174*θ _{d}* = 1.38 × 10

^{−4}

*rad*. 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

*will decrease very slowly with*

_{z}*z.*In the case 2, Δ

*Y*= 0.014

*w*

_{0}= 0.56

*λ*, and Δ

*θ*= −0.96

*θ*= 7.64 × 10

_{d}^{−3}

*rad*. 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>3

*z*

_{0}, it approaches the spot size of Gaussian beam

*w*(

*z*). So, two opposite spin components of transmitted beam are nearly resolvable after 3

*z*

_{0}. However, the spin separations in cases 1 and 2 are smaller than the Gaussian beam size in the region of 0<z<3

*z*

_{0}. In this region, we can maximize Δ

*by optimizing incident angle*

_{z}*θ*and thickness of metamaterial

_{i}*d*. The upper limit of Δ

*is found to be the Gaussian beam size*

_{z}*w*(

*z*). In case 3, we maximize Δ

*for z =*

_{z}*z*

_{0}, we get Δ

*= 0.99*

_{z}*w*(

*z*

_{0}) = 56

*λ*, with

*d*= 0.61

*λ*,

*θ*= 8.66°. The spatial and angular spin splitting are Δ

_{i}*Y*= 0.70

*w*

_{0}, and Δ

*θ*= 0.69

*θ*. Although the spin splitting Δ

_{d}*in this case is smaller than that of case 1 at the beginning, it grows faster, and overtakes at*

_{z}*z*= 0.43

*z*

_{0}. Comparing with the case 2, the spin splitting of case 3 is smaller after

*z*= 2.5

*z*

_{0}, However, in the vicinity of z =

*z*

_{0}, the spin splitting in case 3 are the largest one. At

*z*=

*z*

_{0}plane, the spin splitting of case 1 and 2 are identical. While at

*z*= 2.5

*z*

_{0}plane, the spin splitting of case 2 and 3 are the same.

The spin splitting of transmitted beam is evident in Fig. 6(b), where the intensity distributions of RCP and LCP components in *z* = 0, *z*_{0}, and 2.5*z*_{0} 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* = *z*_{0} 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 (Δ*Y* and Δ*θ*) of transmitted beam through an anisotropic metamaterial are equal to the beam waist *w*_{0} and divergent angle *θ _{d}* of the incident Gaussian beam, respectively. The influences of beam waist and the permittivity of metamaterial on Δ

*Y*and Δ

*θ*for fixed thickness

*d*have been demonstrated. The spin splitting of propagated beam Δ

*is the combination of spatial and angular spin splitting. By optimizing*

_{z}*d*and

*θ*, Δ

_{i}*for a given propagation distance*

_{z}*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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