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Abstract
Metasurfaces have been widely studied for manipulating light fields. In this work, a novel metasurface element is achieved with a high circular polarization amplitude conversion efficiency of 88.5% that creates an opposite phase shift ranging from −180° to 180° between incidence and reflection for different spin components. By arranging the elements according to different requirements, spin-dependent reflection, focusing and scattering are demonstrated. It is also demonstrated that tuning of the Fermi energy is an viable way to active control the circular polarization conversion efficiency and expand the applicable bandwidth. The results open a new route for modifying and designing the wavefront of circular polarized light.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces, the planar counterparts of metamaterials, have attracted significant attention in recent years. In contrast to traditional optical devices based on altering light via reflection, refraction and diffraction, metasurfaces manipulate electromagnetic properties depending on the interaction between the incident light and artificial structures. Thus, by tailoring the material, geometry, and orientation of the metasurface, the phase, polarization and amplitude can be tailored [1–4]. Meanwhile, the spatially inhomogeneous arrangement of the metasurface could build the specific phase distribution to artificially manipulate the wavefront of a light beam and a variety of applications have been demonstrated, such as a planar lens [5, 6], anomalous reflection [7,8], holograms [9,10], coding [11,12] and a vortex beam [13–15].
Recently, one of the most popular ideas in metasurface research is spin optical wavefront control based on Pancharatnam–Berry Phase, which introduces a new degree of freedom of light modulation. By arranging elements with different orientations, researchers have demonstrated arbitrarily shaped wavefronts of circular polarized light. Interestingly, those metasurfaces will generate opposite phase variation and result in different phenomenon for spin components, such as photonic spin Hall effect [16, 17], which has not been concerned properly enough. In addition, by manipulating spin light field, those metasurfaces can realize the interaction between the spin angular momentum associated with circular polarization and the orbital angular momentum associated with the wave vector and phase front, which plays an important role in advanced photonic research [18]. Meanwhile, traditional spin manipulation metasurfaces based on noble metals or dielectric materials cannot be tuned over a broad band after fabrication [19–22]. To achieve dynamic tunability of metasurfaces, many promising methods have been proposed, such as hybridizing metasrufaces with phase change materials [23, 24], ferroelectric materials [25], transparent conductive oxides [26], and graphene [27]. Among them, the integration of metasurfaces with graphene becomes particularly attractive due to the unique optical characteristics of graphene, which can be tuned by varying the Fermi energy through electrostatic doping and chemical doping [28–30]. Although previous works have studied into controlling the wavefront of circular polarized light via graphene, those metasurface still cannot reach high efficiency, which is an important issus in practice [31,32].
In this letter, we introduced a graphene metasurface that enables novel spin-dependent wavefront control with an high amplitude conversion efficiency. First, we design an element to achieve a phase shift of −180° to 180° for right-handed circular polarized (RCP) and left-handed circular polarized (LCP) light with an amplitude conversion efficiency of 88.5% at 10 THz. Then, by arranging the proposed element with different orientations, we demonstrate spin-dependent anomalous reflections. Interestingly, the RCP and LCP light show distinct circular dichroisms with two opposite propagations. Moreover, we also use the element to build a spin-dependent flat lens, which could focus one spin component and scatter the other component, and the focal point can be tuned by changing the phase arrangement. Finally, by exploiting the tunable properties of graphene, it is shown that the metasruface could operate in a broad spectrum with high efficiency.
2. Results and discussion
A schematic of the proposed element is shown in Fig. 1(a). It consists of a bottom gold layer, a SiO2 layer and a top graphene sheet. The geometric parameters are described in the figure caption. The permittivity of SiO2 is set as 2.1, and the permittivity of the gold is defined by the Drude model [33]. The optical conductivity of graphene is given by [34]
where ħ is Planck constant; Ef = 1 eV is the Fermi energy of graphene; is the electron relaxation time associated with the mobility μ= 104 cm2Vs−1 [35] and Fermi velocity vf = 106 ms−1.
Fig. 1 (a) The proposed single element: H1 = 4.5 μm, H2 = 0.8 μm, W1 = 1.6 μm, W2 = 1 μm, W3 = 0.3 μm and the separation distance L = 4 μm in both the x and y directions; (b) Simulation results of the polarization reflection coefficient and the circular polarization reflection coefficient versus the incident wave frequency; (c) Simulation results of phase difference for the x- and y- polarized incident waves versus the incident wave frequency; (d) The phase difference between the incident and reflected light 10 μm above the metasurface and the circular cross-polarization coefficients versus the rotated angle of the proposed element at 10 THz.
In the following, we investigate the proposed element using the finite element method in the software COMSOL Multiphysics. The reflection coefficients rxx, ryy, rxy and ryx versus incident frequency are depicted in Fig. 1(b), and the phases of the reflection coefficients versus incident frequency are shown in Fig. 1(c). We find that the graphene sheet mainly coupling with x polarized incident light at 10 THz (left inset in Fig. 1(c)), and the reflection coefficient rxx is 0.78 with a 180° phase shift (i.e., φxx = 180°). The graphene sheet is nearly not coupling with y polarized incident light with a reflection coefficient of 0.99 at 10 THz (left inset in Fig. 1(c)). The phase shift of the reflection coefficient φyy is nearly zero, which is due to the sum of the round trip propagation phase and the reflection phase at the metal ground plane approaches 360°. In addition, there is no cross-polarization at 10 THz. As a result, the proposed element can be treated as a half-wave plate, which has a 180° phase shift in the x direction. The incident circular polarized light would be converted into another spin component with coefficient of 0.885, and the circular copolarization coefficient is 0.105 at 10 THz. We note that such high efficiency is achieved by adjusting the structural parameters of the graphene element to match the relaxation time of graphene so that rxx could be large, and the specific relation between rxx, the relaxation time and the structural parameters of graphene could be described by the temporal coupled mode theory, as discussed in detail in Ref [34,36].
If we rotate the element by θ, the reflection can be described by a Jones matrix as
where is the reflection matrix decided by the results in Figs. 1(b) and 1(c) at 10 THz (rxx = −0.78 and ryy = 0.99). is the rotation matrix. When the incident wave is RCP or LCP, the reflected light can be expressed asHere, RCP is a right-handed circular polarized wave, and LCP is a left-handed circular polarized wave. As shown in Eqs. (3) and (4), there are two reflection terms. The first term represents the circular co–polarization reflection and the reflection coefficient is 0.105 at 10 THz. For the second term, the reflection circular cross–polarization coefficient can be calculated to be 0.885 10 THz which is consistent with the simulation results. Interestingly, it should be mentioned that the reflection phase shift of the circular cross-polarization e±i2θ in second term varies with the element orientation and has the relationship Δφ = ±2θ for different spin components. This means that we can realize wavefront manipulation of the reflected field by arranging the proposed element in a series with different orientations. Moreover, identical conclusions are obtained in the simulation results in Fig. 1(d). We plot the phase difference between the incident circular polarized light and converted reflected light versus rotated angle 10 μm above the metasurface. The phase difference ranges from −180° to 180° with an opposite tendency for RCP and LCP light. The reflection coefficient of the converted circular polarization varies slightly with the rotation angle.
The above simulation and calculation results show that the proposed element has the ability to convert incident circular polarized light into opposite spin components with a high amplitude conversion efficiency of 0.885 at 10 THz. By changing the orientation of the element, the phase shift of the converted circular polarized light undergoes a continuous variation according to the relationship Δφ = ±2θ for different spin components. Therefore, by arranging the proposed element array, an arbitrary phase distribution can be designed to manipulate the wavefront of spin component.
Using the proposed element, we design a periodically arranged metasurface to produce a spin-dependent abnormal reflection, which is called photonic spin Hall effect (PSHE) [37]. Figure 2(a) provides a schematic of the PSHE metasurface. When a light beam illuminates the metasurface, the reflected field will symmetrically split into LCP and RCP components with opposite reflected angles. We choose 18 elements arranged along the x axis, as shown in Fig. 2(b). The elements are linearly rotated with a step of Δθ = 10°. Accordingly, neighboring elements have an incremental reflected phase of ±20° ranging from −180° to 180° for the RCP and LCP components. Those phase shift profiles are similar to two symmetric blazed gratings: the RCP component is diffracted in the +x direction (blue line in Fig. 2(b)), and the LCP component is diffracted in the −x direction (red line in Fig. 2 (b)). The electric field maps of the reflected spin components are plotted in Fig. 2(d). When the normal incident light is RCP, the reflected light is LCP with high amplitude conversion efficiency of 88.5%, and the reflected angle is −24.6° (right in Fig. 2(d)). Similarly, when the incident light is LCP (left in Fig. 2(d)), the reflected light is mainly RCP, and the reflected angle is 24.6°. This shows a distinct PSHE phenomenon for normal incidence, and the reflected RCP and LCP components assume opposite propagation directions.
Fig. 2 (a) Schematic of the proposed PSHE metasurface; (b) Phase shift profile of the reflected light and rotated angle distribution with 18 elements for one period; (c) Reflected angle versus period length; Electric field maps for the LCP (left) and RCP (right) components with (d) 18 elements, (e) 9 elements and (f) 27 elements for one period at 10 THz.
Moreover, the reflected angle can be controlled by changing the period length. Proposed elements with different orientations have equal reflected amplitudes and phase shifts ranging from −180° to 180° in a period, so the reflected angle can be defined by General Snell’s law as [38]
where θr is the reflected angle, λ is the wavelength, dφ/dx is the phase gradient, and NL is the period length with an element length L and number of elements in a period N. Equation (5) apparently suggests that the reflected angle is controlled by the incident wavelength and the period length. Figure 2(c) shows a calculation of influence of the period length, indicating the reflected angle decreases as the period length increases. To verify this prediction, we also simulate the case of N = 9 with a rotation angle step of Δθ = 20° and N = 27 with a rotation angle step of Δθ = 6.7°, as shown in Figs. 2(e) and 2(f), respectively. These phenomena are consistent with the results in Fig. 2(c) in that the reflected light beam will symmetrically spilt into LCP and RCP components with a reflected angle of ±56.4° for N = 9 and a reflected angle of ±16.1° for N = 27, respectively. The rapidly varying phase shift change (black line in Figs. 2(e) and 2(f)) caused by the decreasing period length leads to an increase in the reflected angle. As a result, the reflection direction is effectively manipulated by adjusting the period length NL.In this manner, a spin-dependent abnormal reflection (PSHE) is realized by the proposed metasurface. The reflected light field symmetrically splits into two different propagation directions for the RCP and LCP components, and the reflected angle can be controlled by changing the period length. Notably, the phenomenon of the traditional PSHE derived from reflection and refraction at an interface is very weak, with a transverse shift of only several wavelengths [39]. In contrast, the PSHE based on the metasurface is actually enhanced by a spin-dependent phase gradient and achieves high efficiency and controllability.
Beside the PSHE metasurface discussed above, in the next we show that metasurface cylindrical lens can also be designed by engineering our proposed elements. Figure 3(a) shows a schematic of the spin-dependent flat lens metasurface. When light beam incidence occurs, the reflected LCP component will be focused, and the reflected RCP component will be scattered. To control the reflected LCP focus at the focal point, the phase shift of the reflected LCP component Δφ(x) is designed as:
where k is the wavevector, f is the focal length, x is the coordinate of the element center on the x axis, and x0 is the x coordinate of the focal point. According to the relationship between rotation angle and phase shift in Eqs. (3) and (4), the distribution of rotation angle can be defined as We choose 37 elements arranged along the x axis, and the orientation distribution is calculated by Eqs. (6) and (7) with x0 = 0 and f = 50 μm. Also, the phase difference for the RCP (red) and LCP (blue) components has been calculated and plotted in Fig. 3(b). Interestingly, the phase shifts of the RCP and LCP components are inverse and exhibit an opposite phase variation, so the metasurface acts like a concave mirror for the RCP component and a convex mirror for the LCP component. Figure 3(c) shows the simulated reflected LCP field maps when the normal incident field is RCP. It is clear that the incident RCP light mainly converts to LCP light and that the reflected LCP light is scattered. Moreover, we also plot the reflected LCP field map when the normal incident field is RCP in Fig. 3(d). The result reveals the reflected LCP component is focused. Furthermore, to define the focal length of such metasurface, we simulated the light intensity of the LCP component in Fig. 3(e). We find the maximum of light intensity (i.e. focal point) is located at 47.0 μm above the metasurface. Such simulated result shows that the focal length has a little difference from the initial design, because 37 elements cannot cover the entire continuous phase shift range from −180° to 180°. Then, we plot the normalized intensity along the x direction through the focal point in Fig. 3(f). The full width at half maximum at the focal point is 15.2 μm which is about half of the incident wavelength (30 μm).Fig. 3 (a) Schematic of the flat lens metasurface; (b) reflected phase shift profile and rotation angle distribution of the proposed metasurface; electric field maps for the (c) RCP and (d) LCP components; (e) Electric field intensity map of the reflected LCP component; (f) Normalized intensity along x direction through focal point at z=47 μm.
Certainly, the spatial tunability of focusing and scattering could also be realized by altering the phase shift profile. As an example, we choose the x coordinate of the focal point to be x0 = 40 μm and the focal length to be f = 70 μm, and the profiles of the phase shift and rotation angle are plotted in Fig. 4(a). Figures 4(b) and 4(c) present the reflected electric field maps of the RCP and LCP components. The reflected field is similar to a special PSHE in that when light is normally incident on the proposed metasurface, the reflected RCP component is scattered and propagates along the −x direction, and the reflected LCP component is focused and propagates along the +x direction. We also plot the intensity map of the LCP component in Fig. 4(d), which illustrates that the focal point is at (x, y)=(36.7 μm, 59.5 μm).
Fig. 4 (a) Reflected phase shift profile and rotation angle distribution of the proposed metasurface; (b) Electric field map for the RCP component; (c) Electric field map for the LCP component; (d) Intensity map of the reflected LCP light.
A spin-dependent flat lens is achieved by arranging 37 elements artificially. We can focus one spin component while scattering the other component. Undoubtedly, by varying the orientation distribution, the position of the focal point could be moved to an arbitrary point. Moreover, a special PSHE has been achieved when the x coordinate of the focal point is not zero.
It is well accepted that graphene metasurface has excellent property for tuning the conductivity through changing Fermi energy [40]. To achieve tunability of the circular polarization wavefront, we simulate the reflection coefficient of the circular cross-polarization as a function of the Fermi energy of graphene and incident frequency, and the result is shown in Fig. 5(a). By changing the Fermi energy from 0.6 eV to 1.2 eV, the reflection coefficient of the circular cross-polarization can be changed from 0 to 0.92 without changing the structural parameters. An increase in the Fermi energy can lead to a blueshift in the maximum efficiency. Therefore, the simulation results clearly demonstrate that the amplitude of the reflected circular polarization can be tuned by the Fermi energy of graphene over a wide frequency range. Furthermore, we can realize broadband performance with high efficiency by tuning the Fermi energy to a suitable value. For example, when the incident frequency is 11 THz, we can tune the Fermi energy of graphene to 1.1 eV for a high amplitude conversion efficiency of 90%. With the same orientation distribution as in Fig. 2(b), the PSHE phenomenon reappears as shown in Fig. 5(b). These elements are linearly arranged with a rotation angle step size of Δθ = 10°, the reflected angle is ±22.3° for the LCP and RCP components, respectively. For the case of a flat lens, we do not need to change the structure of the proposed element, but the phase shift distribution should be reconstituted to maintain a focal length of 50 μm for the LCP component by use of Eqs. (6) and (7). The reflected RCP component is scattered, and the LCP component is focused, as indicated in Figs. 5(c) and 5(d). Above all, broadband spin wavefront manipulation has been realized by tuning the Fermi energy of graphene to a suitable value without changing the structural parameters.
Fig. 5 (a) Circular cross-polarization reflection coefficient as a function of the Fermi energy and the incident wave frequency; (b) PSHE metasurface electric field maps for the LCP (left) and RCP (right) components with 18 elements at 11 THz; (c) Flat lens metasurface electric field map for the LCP component; (d) Flat lens metasurface electric field map for the RCP component.
3. Conclusion
In conclusion, we designed a graphene metasurface element with a high circular polarization amplitude conversion efficiency of 88.5% at 10 THz. The phase of the reflection wave could be tuned by element orientation with the relationship of Δφ = ±2θ. By arranging those graphene elements with different orientations, we could realize spin wavefront control. When the phase shift profiles are arranged like two gratings for RCP and LCP, we achieved the circular dichroism phenomenon called photonic spin Hall effect, and the spin-dependent reflected angle could be tuned by the period length. We also designed a spin-dependent flat lens that focuses one spin component and scatters the other component, and the focal point can be spatially tuned by adjusting the distribution of the element orientations. In addition, by tuning the Fermi energy of graphene, our results can be extended to broadband frequencies without changing the structure of the proposed element. We presented here a possible solution to easily manipulate spin of light by engineering the graphene metasurfaces, and the findings may provide a new opportunity for spin photonic devices.
FUNDING
National key Basic Research Program of China (2014CB340203); National Natural Science Foundation of China (11574308); the Basic Science and Frontier Technology Research Program of Chongqing (cstc2017jcyjAX0038); the CAS Western Light Program 2016.
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