1. Introduction

Controlling EM waves is always fascinating and appears in a wide range of applications. One of the approaches to achieve this aim is typically based on steering the propagation phase of light traveling inside an optical medium, such as dielectric lens. Nevertheless, their utility is tempered by the limited contrast of refractive index available from conventional materials and methods of fabrication. In order to resolve this issue, the application of metamaterials was previously employed, which allow one to design materials with specific and exotic optical properties, including negative refraction [1, 2] and phase holograms [3]. But the relatively large size is not conducive to realize sub-miniaturized and ultra-compact optical devices.

An alternative approach for light control at nanoscale, which seems more feasible for practical realization and implementation, is two-dimensional (2D) metal nanostructures with subwavelength periodicity, also known as metasurface. Metasurfaces, a new class of metamaterials that consist of only a monolayer of planar metallic structures, have shown great promise for achieving full control of the wave front of light with low fabrication cost as they do not require complicated three-dimensional (3D) nanofabrication techniques. The research field about metasurface is rapidly expanding recently, and have already been used to rotate polarization [4], engineer the optical spin–orbital interaction [5, 6], create optical vortex beams [7, 8], couple propagating waves to surface waves [9], create ultra-thin planar lenses [10–14], and fabricate phase holograms [15]. Furthermore, such an approach has also been extended to effectively manipulate both amplitude and phase of incident light [16]. By using and properly arranging the metasurface units in which the structural parameters vary in a subwavelength scale, and the incident light onto an ultrathin metasurface can be bent by an angle using the phase gradient across the metasurface. And the bent angle decided by the generalized Snell's law [7, 8, 17, 18]. However, it should be noted that only a limited phase range, from 0 to $\pi$, could be reached for co-polarized transmitted or reflected light, which is not enough to fully control the wavefront of incident light [19, 20]. The issue has been resolved by introducing cross-polarized transmitted or reflected light, in which the whole phase variation of 2$\pi$ can be realized [21–23]. In order to obtain the required phase shifts while maintaining constant amplitudes, here, we use the metal rectangular split-ring resonator to build the block units that possess the same total length. So far, almost all studies on metasurfaces have focused on manipulating linearly polarized light. In contrast, only a few of them [24, 25] have been devoted to employing metasurfaces to control circularly polarized light.

In this paper, we propose an ultra-thin plasmonic metasurface constructed by the metal rectangular split-ring resonators (MRSRR) array and demonstrate theoretically and numerically that cross-polarized transmitted light that is the transmitted light with an opposite helicity compared to the incident light, can be obtained and bended to $\pm {23}^{°}$with a circularly polarized light incidence at 808nm. Nevertheless, when the linearly polarized light is used as the incident light, the incident linearly polarized light will be decomposed into both right-handed circular polarization (RCP) and left-handed circular polarization (LCP) with opposite directions.

2. Theoretical formulation

To achieve the required phase coverage of 0~2$\pi$ while maintaining constant scattering amplitudes, it is just need to modify the wavefront with phase discontinuities by rotating optical axes of resonators. Here we assume that the incident wave is a plane wave at normal incidence on the metasurface. The metasurface is compose of the MRSRR array as shown in Fig. 1(a), in which the angle of optical axes are different. Under normal incidence, we can define the general transmission matrix ${\underset{¯}{T}}_{CP}$, describing the complex transmission coefficient for circularly polarized light [20],

 

Fig. 1 (a) Schematic super unit cell of the plasmonic metasurface which exhibits equal amplitude and linear phase gradient for demonstrating the generalized laws of refraction. (b) The basic unit cell with P = 250nm, L = 150nm, w = 30nm, various L₁, L₂ and the thickness of resonator is t = 100nm.

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where the${{\displaystyle t}}_{xy}$represents the complex transmission coefficient of the transmitted light, linearly polarized in $x$ direction for excitation in the $y$ direction, as well as $t_{yx}$, $t_{xx}$, ${{\displaystyle t}}_{yy}$ , and $T_{LR}$denotes the transmission coefficient for LCP by RCP illumination, with similar definitions for all the other elements.

For a unit of MRSRR, the optical axis rotated by angle $\theta$ in the anticlockwise direction, and the corresponding transmission coefficients can be represented using the Jones matrix $T_{CP}^{*}$ [26],

A LCP plane wave can be described by the Jones electric-field vector as E_in = (1 i)^T, and after passing through the metasurface, the transmitted field can be expressed by the following form:

In Eq. (3), the first term denotes the same polarization as the incident polarized light and the second term represents the cross-polarized light which possesses an additional phase$2\theta$. Therefore, we can control the wavefront of cross-polarized light (here, it is RCP), just by changing the angle of the optical axes of resonators. And the bend angle of transmitted light is determined by gradient phase shifts related to the generalized Snell’s law [7, 8, 17, 18] that can be expressed as follow:

where ${{\displaystyle \theta }}_i$ and ${{\displaystyle \theta }}_t$ are the incident angle and refractive angle, respectively. ${{\displaystyle n}}_i$(${{\displaystyle n}}_t$) is the refractive index in the incident (refracted) region. ${{\displaystyle \lambda }}_0$ is the incident wavelength in vacuum, and$d\Phi /dx=2\theta =2\pi /D$ (D is the period of super unit cell).

In our case, the incident and refractive region are glass (${{\displaystyle n}}_i$ = 1.5) and vacuum (${{\displaystyle n}}_t$ = 1), respectively. And the refractive index of gold is described by an interpolation of experimental data [27]. And the incident light normally illuminates the metasurface from the glass substrate side, therefore, the refraction angle is determined by

In order to precisely control each optical axis of MRSRR units as depicted in Fig. 1(b), the two branches' sizes (L₁ and L₂) of MRSRR units can be considered as two variable degrees of freedom (DOF). As long as the size of L₁ and L₂ are select reasonably, the relative optical axis' direction can be controlled precisely. As illustrated in Figs. 2(a) and 2(b), the amplitude and phase of the transmitted cross-polarized light can be efficiently controlled by exploiting two DOF in the nanostructure geometry. This particular feature of our metasurface is highly desired to modulate the wavefront of light as it has the linear phase relationship and it is robust against fabrication tolerances and variation of metal properties because of the much simpler structure geometry. We chose four antennas, as indicated by stars in Figs. 2(a) and 2(b), which provide an incremental phase of$\pi /4$from left to right for the cross-polarized light. By simply rotating the structures by 90°, the additional$\pi$ phase shift can be obtained for cross-polarized light [8, 21]. Therefore, the phase shift across the whole super unit cell constituted of eight antennas as depicted in Fig. 1(a) could cover a broad range of 0~2$\pi$. Full-wave simulations for a wavelength of 808nm confirm that the amplitudes of the cross-polarized light scattered by the eight antennas are nearly equal, with nearly linear phases (as shown in Fig. 3).

 

Fig. 2 The simulated amplitude (a) and phase shift (b) of the cross-polarized light for the MRSRR in Fig. 1(b) with various lengths L₁ and L₂ , for the LCP normal incidence at wavelength of 808nm. The four stars in (a) and (b) indicate the corresponding used values of L₁ and L₂.

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Fig. 3 The simulated cross-polarized transmittance and corresponding phase shift when each individual resonator is used for the LCP normal incidence, which demonstrates that the super unit cell possesses constant amplitude and linear phase shift.

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3. Result and discussion

Under the normal incidence, the anomalous refraction phenomenon is observed for both LCP and RCP. Full-wave simulations were performed for both LCP and RCP light at a wavelength of 808nm. Figures 4(a) and 4(b) illustrate the transmitted cross-polarized electric field distributions of RCP and LCP by LCP and RCP normal incidence respectively, which show the opposite direction of deflection. The bending angles, as shown in Figs. 4(c) and 4(d), are$-{23}^{°}$and${23}^{°}$for LCP and RCP incidence, respectively, which are agree well with the calculated bending angles of$\pm {23.83}^{°}$deduced from Eq. (5). As depicted in Fig. 3, the transmission amplitude $\left|t\right|$for cross-polarized light is about$33\%$ at the wavelength of 808nm. The low transmission amplitude for cross-polarized light results from the existence of reflection. As discussed above, the bending angle is related to the helicity of circular polarization for which the deflecting direction is opposite for LCP and RCP. It should be noted that the metasurface exhibits negative refraction for both LCP and RCP normal incidence.

 

Fig. 4 The transmitted cross-polarized electric field distributions for the LCP (a) and RCP (b) incidence. (c and d) The normalized transmission far-field scattered light (cross-polarized light) as a function of angles for the LCP (c) and RCP (d) at normal incidence. And the simulated bending angles agree well with the generalized Snell’s law. The blue and red arrow (line) in both (c) and (d) represents RCP and LCP respectively.

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It is quite intriguing to note that if the normal incident light is X-polarized light at the wavelength of 808nm, as shown in Fig. 5(a), the transmitted light will consist of three components, i.e. the normal refracted light, the RCP component with bending angle of $-{23}^{°}$and LCP component with${23}^{°}$. This phenomenon can be attributed to the fact that linearly polarized light can be decomposed into circularly polarized light with opposite helicity. And, the anomalous refractions with different directions occur for the different incident circularly polarized lights. Therefore, X-polarized light will be decomposed into two different kinds of circular polarizations of LCP and RCP, which deflect with opposite bending angles. As any linear polarization state can be decomposed into LCP and RCP, it is expected that two circular polarization states can be achieved simultaneously for arbitrary linearly polarization incidence because of the anomalous refraction, which can be used for spin selective image by LCP and RCP.

 

Fig. 5 (a) Normalized transmission curves indicate that transmitted light consist of normal refraction, RCP and LCP. Black, red and blue line represent the normal refraction, LCP and RCP. (b) The cross-polarized transmittance as a function of frequency and angle. The dashed curve is the theoretically calculated frequency-dependent bending angle.

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To further demonstrate the robustness of our metasurface, we plotted the cross-polarized transmittance as a function of frequency and transmission angle (the co-polarized transmission is negligible) under LCP incidence. As depicted in Fig. 5(b), the anomalous beam is transmitted at a frequency-dependent refraction angle$\theta$_t following the Eq. (5) which indicate that our metasurface is robust over abroad bandwidth. The calculated frequency-dependent anomalous refraction angle is plotted as the dashed curve in Fig. 5(b), revealing excellent agreement with the simulation results.

Additional numerical simulations show that anomalous refraction can also be obtained by using the same method in infrared wavelength ranges. we modulate the geometric parameters of MRSRR suitable for the wavelength of 1200nm as P = 500nm, L = 300nm, w = 50nm and corresponding variables of L₁ and L₂. We investigate the bending phenomenon for both LCP and RCP under normal incidence at the wavelength of 1200nm. As depicted in Fig. 6(a), the anomalous transmission light bends to left under LCP incidence. For RCP incidence, the transmission light bends to right, as shown in Fig. 6(b), which agrees well with the theoretical expectation. And the bending angle is also related to the incident circular polarization which further demonstrates the correctness of our scheme. The use of bigger super cell results in a larger transmission energy which the transmission amplitude$\left|t\right|$ is about$38\%$. And the bending angle is$\pm {17}^{°}$as shown in Fig. 6(c), which is smaller (compared with Figs. 4(c)-4(d)). According to Eq. (4), this phenomenon should be attributed to the variety of wavelength and phase shift $d\Phi /dx$which associate with the period of the super cell, i.e. $d\Phi /dx=2\pi /D$. When the X-polarized normal-incidence light (the wavelength of 1200nm) irradiates on the MRSRR array, as shown in Fig. 6(d), there are also three components i.e. the normal refraction, the RCP component with bending angle of $-{17}^{°}$and LCP component with${17}^{°}$_.

 

Fig. 6 (a and b) indicate the electric field distributions of cross-polarized light for the left-handed circular polarization incidence (a) and right-handed circular polarization normal incidence (b) which indicates that the wavefront across the metasurface is reconstructed. (c) The bending angle of cross-polarized light for both LCP and RCP whose directions are opposite. (d) The transmitted light consist of three components i.e. the normal refraction, the RCP component with bending angle of $-{17}^{°}$ and LCP component with ${17}^{°}$, under normal incidence with X-polarized light, which denoted by black, blue and red line, respectively.

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As the angle of anomalous refraction is proportional to the incident wavelength and phase shift$d\Phi /dx$, it is expected that the broadband angle range can be achieved by simply adjusting the incident wavelength and phase shift. Especially, the required phase shift can be achieved by adjusting configuration parameter, such as the period of unit cell P, the width of aperture, and L₁, L₂ . Therefore, our scheme can be exploited for flexibly controlling the propagation direction of circularly polarized light which has evoked enormous interest recently.

4. Conclusion

In summary, we have proposed an ultra-thin plasmonic metasurface with phase discontinuity constructed by anisotropic MRSRR array for LCP and RCP wavefront manipulation. The phase discontinuity can be achieved by precisely controlling the optical-axis rotation profile of the resonators in a subwavelength scale, in which the optical-axis rotation is related to the reasonable control of two DOF in our nanostructure. By employing the metasurface, an incident circularly polarized light can be converted into a deflected cross-polarized light, and the normal-incidence linearly polarized light can be decomposed into LCP and RCP lights. These characters of the designed MRSRR array are significant for the circular polarization analyzer and the spin selective image by LCP and RCP.