Polarization Controlled Dual Functional Reflective Planar Metalens in Near Infrared Regime

The metalens has been a hotspot in scientific communications in recent years. The polarization-controlled functional metalens is appealing in metalens investigation. We propose a metalens with dual functions that are controlled by polarization states. In the first design, when applied with x- and y-polarized light, two focal spots with different focal lengths are acquired, respectively. The proposed metalens performs well when illuminated with adjacent wavelengths. In the second design, the reflected light is focused when applied with x-polarized light, and when applied with y-polarized light, the reflected light is split into two oblique paths. We believe that the results will provide a new method in light manipulation.

In this paper, a metal-insulator-metal (MIM) structure is proposed to achieve the polarization controlled dual functional two-dimentional (2D) cylindrical metalens, which consists of rectangle gold antennas and a gold mirror spaced by a dielectric layer. The proposed metalens possesses two functions that can be realized when illuminated with x-and y-polarized light. The x-direction and y-direction length of the rectangle gold antenna are adjusted to control the phase and reflectance of the reflected light, because the length of the dipole resonances along x-and y-directions are determined by the x-direction and y-direction lengths, respectively. We first investigate the influence of the y-direction length or x-direction length on the dipole along the x or y-direction. By presenting the phase and reflectance of the reflected light for x-polarization incidence with different x-direction and y-direction lengths of the rectangle antenna, we find that the y-direction length of the rectangle has little influence on the phase and reflectance in most of its length range. Then we apply the phase approach to design the polarization-controlled metalens, in which the y-direction or x-direction length is fixed as 100 nm when varying the x-direction or y-direction length. We design a metalens with the focal length of F = 5 µm and F = 5 µm when applied with x-and y-polarized light, which focus well as desired. Then we validate the approach by presenting two metalenses with focal length of 5 and 15 µm for x-and y-polarization incidence only. The y-direction and x-direction lengths are fixed as 100 nm. The focusing effects agree well with that of the proposed polarization-controlled metalens. The proposed metalens works well within a broadband of wavelengths that range from 750 to 850 nm. We also design the metalens to split y-polarized normal incident light and focus the x-polarized light. Therefore, the dual functional metalens can be achieved by tuning the incident polarization state. Figure 1 illustrates the schematic of the proposed 2D cylindrical lens, which is a metal-insulator-metal structure to form a Fabry-Perot cavity to enhance the interaction between light and resonance antenna. The resonance antenna and the bottom mirror are selected as Au with data acquired from [33]. The dielectric spacer is chosen as MgF 2 with a refractive index of 1.892 [14]. The thicknesses of Au antenna, MgF 2 spacer and Au mirror shown in Figure 1b are set as t a = 30 nm, t d = 50 nm and t s = 130 nm, respectively. The top view of the unit cell of the metalens is shown in Figure 1c. The period in x-and y-directions are p x = p y = 200 nm. The resonance antenna is a rectangle, which possesses dipole resonances along both the x-and y-directions. The rectangle lengths along x-and y-directions are variable to control the phase and reflectance of the reflected light for x-and y-polarization incidences, respectively. Thus, we can separately control the wavefront of the reflected x-and y-polarized light by configuring the rectangle lengths along x-and y-directions. The working mechanism of the proposed metalens is shown in Figure 1a, where different polarization state leads to different function. For numerical analysis, all simulations are carried out by using the finite-difference time-domain software (Lumerical FDTD solutions 8.15.736.0). PML boundary condition is applied in zand x-directions. Periodic boundary condition is applied in the y-direction. The minimal mesh size is 4 nm.

Design and Simulation Method
Coatings 2020, 10, x FOR PEER REVIEW 2 of 9 and reflectance of the reflected light for x-polarization incidence with different x-direction and ydirection lengths of the rectangle antenna, we find that the y-direction length of the rectangle has little influence on the phase and reflectance in most of its length range. Then we apply the phase approach to design the polarization-controlled metalens, in which the y-direction or x-direction length is fixed as 100 nm when varying the x-direction or y-direction length. We design a metalens with the focal length of F = 5 μm and F = 5 μm when applied with x-and y-polarized light, which focus well as desired. Then we validate the approach by presenting two metalenses with focal length of 5 and 15 μm for x-and y-polarization incidence only. The y-direction and x-direction lengths are fixed as 100 nm. The focusing effects agree well with that of the proposed polarization-controlled metalens. The proposed metalens works well within a broadband of wavelengths that range from 750 to 850 nm. We also design the metalens to split y-polarized normal incident light and focus the x-polarized light. Therefore, the dual functional metalens can be achieved by tuning the incident polarization state. Figure 1 illustrates the schematic of the proposed 2D cylindrical lens, which is a metal-insulatormetal structure to form a Fabry-Perot cavity to enhance the interaction between light and resonance antenna. The resonance antenna and the bottom mirror are selected as Au with data acquired from [33]. The dielectric spacer is chosen as MgF2 with a refractive index of 1.892 [14]. The thicknesses of Au antenna, MgF2 spacer and Au mirror shown in Figure 1b    Generally, the influence of the y-direction length b or x-direction length a on the dipole along the x or y-direction cannot be ignored, which results in amplitude and phase deviations for the x-or y-polarization incidences. To explore this, we choose the working wavelength as 800 nm. We show the phase and reflectance of the reflected light for x-polarization incidence in Figure 2 with different lengths b and a of the rectangle antenna. A near 2π phase shift can be acquired when the length a increases from 10 to 190 nm, and the length b has little influence on it in most of the length range as shown in Figure 2a,b. Due to the symmetry property of the structure, an identical performance can be acquired for y-polarization incidence.

Design and Simulation Method
Coatings 2020, 10, x FOR PEER REVIEW 3 of 9 Generally, the influence of the y-direction length b or x-direction length a on the dipole along the x or y-direction cannot be ignored, which results in amplitude and phase deviations for the x-or y-polarization incidences. To explore this, we choose the working wavelength as 800 nm. We show the phase and reflectance of the reflected light for x-polarization incidence in Figure 2 with different lengths b and a of the rectangle antenna. A near 2π phase shift can be acquired when the length a increases from 10 to 190 nm, and the length b has little influence on it in most of the length range as shown in Figure 2a,b. Due to the symmetry property of the structure, an identical performance can be acquired for y-polarization incidence.  Figure 2c shows the phase and reflectance of the reflected light for x-polarization incidence when x-direction length a increases from 10 to 190 nm and the y-direction length b is fixed as 100 nm. Figure 2d shows the phase and reflectance of the reflected light for y-polarization incidence when y-direction length b increases from 10 to 190 nm and the x-direction length a is fixed as 100 nm. The insets show the z component electric field distributions at the resonant length. Two dipole resonances along xand y-directions are acquired, together with the Fabry-Perot resonance, a near 2π phase shift can be achieved. We use this approach to design the metalens that possesses different functions controlled by the polarization states of the incident light.

Results and Discussion
As illustrated above, we utilize the phase shift in Figure 2c,d to design the focusing metalens, which possesses the focal length of 5 μm F = for x-polarization incidence and 15 μm F = for ypolarization incidence. To design a metalens to focus incident light, the phase profile of the metalens should follow the expression [34]:  Figure 2c shows the phase and reflectance of the reflected light for x-polarization incidence when x-direction length a increases from 10 to 190 nm and the y-direction length b is fixed as 100 nm. Figure 2d shows the phase and reflectance of the reflected light for y-polarization incidence when y-direction length b increases from 10 to 190 nm and the x-direction length a is fixed as 100 nm. The insets show the z component electric field distributions at the resonant length. Two dipole resonances along x-and y-directions are acquired, together with the Fabry-Perot resonance, a near 2π phase shift can be achieved. We use this approach to design the metalens that possesses different functions controlled by the polarization states of the incident light.

Results and Discussion
As illustrated above, we utilize the phase shift in Figure 2c,d to design the focusing metalens, which possesses the focal length of F = 5 µm for x-polarization incidence and F = 15 µm for y-polarization incidence. To design a metalens to focus incident light, the phase profile of the metalens should follow the expression [34]: where x is the horizontal position from the center of the metalens, ∆x is the horizontal shift of the focal point, λ 0 is the incident wavelength, and F is the focal length. Based on Equation (1), we calculate the phase distribution curves for F = 5 µm and F = 15 µm, and show them in Figure 3a,b. The corresponding length distributions of x-direction and y-direction lengths a and b are shown in Figure 3c,d, according to which we design and simulate the planar metalens with 60 unit cells with x-and y-polarization incidences, respectively. The electric field intensity distributions for x-and y-polarization incidences are shown in Figure 3e Thus, the metalens with two polarization-controlled focal points is achieved. In practice, a minor inclination of polarization plane is inevitable, which results in a negligible focusing effect for another polarization state.
Coatings 2020, 10, x FOR PEER REVIEW 4 of 9 where x is the horizontal position from the center of the metalens, x  is the horizontal shift of the focal point, 0  is the incident wavelength, and F is the focal length. Based on Equation (1), we calculate the phase distribution curves for 5 μm F = and 15 μm F = , and show them in Figure 3a,b.
The corresponding length distributions of x-direction and y-direction lengths a and b are shown in Figure 3c,d, according to which we design and simulate the planar metalens with 60 unit cells with x-and y-polarization incidences, respectively. The electric field intensity distributions for x-and ypolarization incidences are shown in Figure 3e,f, respectively. The reflected lights are well focused in the air side except for a little deviation in the focal length (4.95 and 13.32 mm for x-and y-polarization incidences) from the theoretical values. Because the phase shift resulting from the length change cannot cover the full 2π range, we select the adjacent unit cell instead as an approach. The longitudinal and transverse sizes of the focal spots are represented by full width at half maximum (FWHM) of the focal spots along x-and z-directions. For x-polarization incidence, the FWHM values along x-and z-directions are 0.78 and 2.72 mm. For y-polarization incidence, The FWHM values are 1.1 and 9.2 mm, respectively. The focusing efficiencies, defined as the proportion of the incident light energy going to the central focal spot, are calculated to be 41% and 45% for x-and y-polarization incidences. Thus, the metalens with two polarization-controlled focal points is achieved. In practice, a minor inclination of polarization plane is inevitable, which results in a negligible focusing effect for another polarization state. To investigate the influence of the x-direction length a and y-direction length b on the focusing effect of y-and x-polarization incidences, we design two metalenses that focus x-polarized light and y-polarized light only. For x-polarization incidence, only the x-direction length a is designed for focusing with a focal length of 5 μm F = as shown in Figure 4a, while the y-direction length b is fixed as 100 nm as shown in Figure 4c. For y-polarized light focusing, x-direction length a is fixed as 100 nm as shown in Figure 4b, while the y-direction length b is designed for focusing with a focal length of 15 μm F = as shown in Figure 4d. We simulate the two metalenses with x-and ypolarization incidences, and the results are shown in Figure 4e,f, from which we can see that the results are nearly the same as that shown in Figure 3e,f. Therefore, the influence of the x-direction To investigate the influence of the x-direction length a and y-direction length b on the focusing effect of y-and x-polarization incidences, we design two metalenses that focus x-polarized light and y-polarized light only. For x-polarization incidence, only the x-direction length a is designed for focusing with a focal length of F = 5 µm as shown in Figure 4a, while the y-direction length b is fixed as 100 nm as shown in Figure 4c. For y-polarized light focusing, x-direction length a is fixed as 100 nm as shown in Figure 4b, while the y-direction length b is designed for focusing with a focal length of F = 15 µm as shown in Figure 4d. We simulate the two metalenses with x-and y-polarization incidences, and the results are shown in Figure 4e,f, from which we can see that the results are nearly the same as that shown in Figure 3e,f. Therefore, the influence of the x-direction length a and y-direction length b on the focusing effects of y-and x-polarization incidences are negligible.
Coatings 2020, 10, x FOR PEER REVIEW 5 of 9 length a and y-direction length b on the focusing effects of y-and x-polarization incidences are negligible. In addition, we also investigate the cases of the proposed metalens working in other incident wavelengths. The simulated results are shown in Figure 5. Figure 5a,b show the electric field distributions when illuminated with 750 nm wavelength x-and y-polarized light, respectively. Figure 5c,d show that of 850 nm wavelength incident light. From Figure 5 we can conclude that the proposed metalens can work well within a broadband wavelength that ranges from 750 to 850 nm. In addition, we investigate the focusing effects for other incident wavelengths. The focal lengths for different incident wavelengths are listed in Figure 5e,f, from which we can see that the focal length changes when incident light wavelength is changing, and the focal length is inversely proportional to the wavelength of light. In addition, we also investigate the cases of the proposed metalens working in other incident wavelengths. The simulated results are shown in Figure 5. Figure 5a,b show the electric field distributions when illuminated with 750 nm wavelength x-and y-polarized light, respectively. Figure 5c,d show that of 850 nm wavelength incident light. From Figure 5 we can conclude that the proposed metalens can work well within a broadband wavelength that ranges from 750 to 850 nm. In addition, we investigate the focusing effects for other incident wavelengths. The focal lengths for different incident wavelengths are listed in Figure 5e,f, from which we can see that the focal length changes when incident light wavelength is changing, and the focal length is inversely proportional to the wavelength of light. Furthermore, another function called the beam splitter can be considered in the dual functional metalens design, which manipulates the normal incident light into two opposite oblique paths [35]. To bend the normal incident light to an oblique reflected light, a phase gradient should be introduced along the interface of the structure following the generalized Snell's law [36]. For normal incidence, the reflected angle can be expressed by: where 0  , x ,  and i n are the incident wavelength, surface length, phase shift and refractive index of the incident side material, respectively. In our design, the reflection angle of the oblique beams are 30  , then the calculated phase shift between two adjacent unit is 45 . The phase profile calculated from Equation (2) is shown in Figure 6b. The corresponding y-direction length b distribution is shown in Figure 6d, based on which the normal incident y-polarized light can be split.
However, the reflection angle is less than 30 , because the phase shift produced by the gold antennas cannot cover the full 2π range, and we take some approximations in the designing process. Then for x-polarization incidence, the x-direction length a is designed according to Figure 6a,c, which focus the x-polarized incident light with the focal length of 5 μm  Furthermore, another function called the beam splitter can be considered in the dual functional metalens design, which manipulates the normal incident light into two opposite oblique paths [35]. To bend the normal incident light to an oblique reflected light, a phase gradient should be introduced along the interface of the structure following the generalized Snell's law [36]. For normal incidence, the reflected angle can be expressed by: where λ 0 , x, ϕ and n i are the incident wavelength, surface length, phase shift and refractive index of the incident side material, respectively. In our design, the reflection angle of the oblique beams are ±30 • , then the calculated phase shift between two adjacent unit is 45 • . The phase profile calculated from Equation (2) is shown in Figure 6b. The corresponding y-direction length b distribution is shown in Figure 6d, based on which the normal incident y-polarized light can be split. However, the reflection angle is less than 30 • , because the phase shift produced by the gold antennas cannot cover the full 2π range, and we take some approximations in the designing process. Then for x-polarization incidence, the x-direction length a is designed according to Figure 6a,c, which focus the x-polarized incident light with the focal length of F = 5 µm. The simulated results of both x-and y-polarization incidences are shown in Figure 6e,f. For x-polarization incidence, the reflected light is well focused at the point of (x = 0 z = 5 µm), and the y-polarized light is split. The advantage of our results is listed in Table 1 to compare with other works.

Conclusion
In summary, we proposed a dual functional metalens controlled by polarization state of the incident light. Two functions can be achieved by adjusting the incident light with two polarization states. The proposed metalens is designed according to the approach that the dipole resonance is not influenced by the width of the rectangle gold antenna. A metalens working at 800 nm wavelength with focal lengths of 5 μm F = for x-polarization and 15 μm F = for y-polarization incidence is designed, which agree well with the two exact metalens that focus x-and y-polarized light only. The proposed metalens works well within a broadband wavelength that ranges from 750 to 850 nm. We also designed a metalens with two functions of focusing and splitting controlled by incident polarization states. Therefore, the proposed metalens can achieve two functions by applying two orthogonal polarized lights.

Conflicts of Interest:
The authors declare no conflict of interest. The advantage of our results is listed in Table 1 to compare with other works. Table 1. The compare of this work to previous works.

Conclusions
In summary, we proposed a dual functional metalens controlled by polarization state of the incident light. Two functions can be achieved by adjusting the incident light with two polarization states. The proposed metalens is designed according to the approach that the dipole resonance is not influenced by the width of the rectangle gold antenna. A metalens working at 800 nm wavelength with focal lengths of F = 5 µm for x-polarization and F = 15 µm for y-polarization incidence is designed, which agree well with the two exact metalens that focus x-and y-polarized light only. The proposed metalens works well within a broadband wavelength that ranges from 750 to 850 nm. We also designed a metalens with two functions of focusing and splitting controlled by incident polarization states. Therefore, the proposed metalens can achieve two functions by applying two orthogonal polarized lights.

Conflicts of Interest:
The authors declare no conflict of interest.