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
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 . The later one is the so called Imbert–Fedorov (IF) shifts , 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 . 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 . By choosing the eigenpolarization appropriately, a splitting up to 10 wavelengths has been obtained experimentally . 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 . 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  or a thin epsilon-near-zero slab . 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 . 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 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//, ], where ε// is determined by the permittivity of background dielectric, while 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 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 . According to Ref [1,17], the transmitted angular spectrum is given by17]. 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 is10]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 haveEqs. (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, , so the term in the denominator of Eq. (5) can be neglected. Therefore, the maximum spatial spin splitting is approximately obtained when the conditions: 1) , and 2) φ = 0 or π are satisfied simultaneously. These two conditions ensure and |cosφ| reaching their maximum values, respectively. Under these conditions, the spatial spin splitting of transmitted beam is given by
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, = 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 changes rapidly, which results in the fast change of phase parameter φ.
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, 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 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 . 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 . 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.
Most metamaterials are absorptive medium in practical . Here, we set the perpendicular component of permittivity as a complex . The imagine part of indicates loss. A slightly loss <0.001 has little influence on both the normalized spatial (ΔY/w0) and normalized angular (Δθ/θd) spin splitting. For a larger loss with = 0.003, ΔY/w0 and Δθ/θd decrease, as shown in Figs. 4(a) and 4(b). When the real and image parts of are equal, = = 0.01, the image part of epsilon will give strong influence to the transmission coefficient , 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 and . Specially, φ is approximately equal to the phase difference between and when , which holds for incident angle θi>5.74°. Since the phase difference can be modulated by the material loss , the phase φ will vary with . Therefore, when increases from 0 to 0.01, both the spatial and angular spin splitting change signs , as shown by Figs. 4(a) and 4(b). When increases further, the absorption becomes the dominated effect. The amplitude and phase of 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 , the positions of the peaks of both spatial and angular spin splitting increase gradually, which is caused by the increase of .
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 . As shown in Figs. 5(a) and 5(b), when 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 = 0.05 and 0.1 reach 0.99, which is larger than that for = 0.01 even though the thickness of metamaterial is optimized for = 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.
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+-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.
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.
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].
National Natural Science Foundation of China (61675092); Natural Science Foundation of Guangdong Province (2016A030313).
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