Metamaterials have many unusual phenomena, including negative refractive index, giant chirality, and indefinite permittivity^[1-4], which made them widely investigated in recent years. Metasurfaces, the two dimensional versions of metamaterials, have been demonstrated to be a novel approach to control electromagnetic wavefronts^[5-6]. Since Capasso et al. first investigated beam steering in 2011^[7], many kinds of structures have been reported on metasurfaces to achieve the functions of focusing lens^[8-10] and anomalous reflection and refraction^[11]. Among many kinds of metasurfaces, metalens can achieve the function of focusing and anomalous reflection by configuring different materials and shapes of units. Lots of models like nanoslit^[12], nanohole^[13-14], graphene ribbons^[15-19] are introduced in metalens design, where 2π phase shift resulted by the designed antennas is needed in controlling the wavefront. Although much structures have been proposed in past reports, the function of metalens have not been fully investigated.

In this paper, we proposed a new model based on Au and MgF₂, which is independence of polarization^[20-21]. x-polarized and y-polarized incidence are both have the same focusing effect. The proposed metalens can work well in the range of 700 - 850 nm for one-focus, and by configuring the resonance antenna, the proposed metalens can also achieves dual focus and three focus. When we use the incidence of 800 nm wavelength, the focal length of one focus, dual focus and three focus are 9.6, 6.6 and 4.7 μm, respectively. The focal length will decrease as the focus number increases. We believe that our findings are beneficial in designing new function controlling devices.

Figure 1 illustrates the schematic of the proposed 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 reference [22]. The dielectric spacer is chosen as MgF₂ with a refrative index of 1.892^[23]. The thicknesses of Au antenna, MgF₂ spacer and Au mirror shown in Fig.1(b) 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 Fig.1(c). The period in x- and y-directions are p_x=p_v=200 nm. The resonance antenna is a circle, which is independent of different polarized light. Thus, we can control the wavefront of the plane wave regardless of x-polarization or y-polarization incidences. The working mechanism of the proposed metalens is shown in Fig.1(a). For numerical analysis, all simulations are carried out by using the finite-difference time-domain (FDTD) software.

Generally, the resonant antenna is a circle, it has the same effect on different polarized light. To explore this, we choose the working wavelength as 800 nm. It shows the phase shift and reflectance of the reflected light for x-polarization incidence in Fig.2(a) and y-polarization incidence in Fig.2(b). A near 2π phase shift can be acquired when the radius r increases from 10 nm to 90 nm. As we can see, the change in the radius of the resonant antenna has the same effect on x-polarization incidence and y-polarization incidence.

To design metalens to focus incident light, the phase profile of the metalens should follow the expression^[24]:

where x is the horizontal position from the center of metalens; ${\lambda _0}$is the wavelength of the incident; F is the focal length. Based on Eq.(1), we calculated the phase distribution curves for F=10 μm, and show them in Fig.3(a).

According to the expression, we selected the radius of the resonant antenna from Fig.2(a) and Fig.2(b) to design metalens. The focal length is designed as 10 μm and the size of the metalens is designed as 12 μm (60 units). The relationship between the position x and the radius of each unit for x-polarization and y-polarization are shown in Fig.3(a) and Fig.3(d) respectively. We simulated the metalens with 800 nm wavelength of the incident light. The Poynting vector distributions for incidence for x-polarization and y-polarization are shown in Fig.3(b) and Fig.3(e) respectively, (d) and the simulated focal lengths are both 9.6 μm, which agree well with the designed values. The intensity distributions along x-direction for x-polarization and y-polarization are shown in Fig.3(c) and Fig.3(f) respectively and the FWHM (Full Wave at Half Maximum) values for the focal lengths are both 0.67 μm. The results show that x-polarization and y-polarization have exactly the same focusing effect. The small deviations among the simulated focus length and the designed value are because the phase shift does not cover 2π, and we made the approximation when we selected the radius of the resonant antenna. On the other hand, the insufficient mesh accuracy may lead to deviations.

Then we simulated the proposed metalens in other incident wavelengths for x-polarization, thats 650, 700, 750, 850 and 900 nm. Simulated results are shown in Fig.4 and Fig.5. The poynting vector distributions for 650, 700, 750 , 850 and 900 nm are shown in Fig.4(a)−(e), respectively. The intensity distributions for 650, 700, 750, 850 and 900 nm along x-direction are shown in Fig.5(a)−(e), respectively. As we can see, the proposed metalens can works well within a broadband wavelength that ranges of 700−850 nm, and it performs best in the 750−800 nm range.

Further more, the proposed metalens can achieve dual focus by configuring the resonance antenna according to the expression:

where ɑ(x) is the nearest amplitude, d is the distance between the two focals and ɑ₁=ɑ₂=0.5. We designed two focus locate at x=−3, x=3 and the focal lengths are both 10 μm. The relationship between the position x and the radius of each unit is shown in Fig.6(a). We simulated the metalens with 800 nm wavelength of the incident light. The Poynting vector distributions for incidence and the intensity distributions along x-direction are shown in Fig.6(b) and Fig.6(c), respectively. The simulated focal lengths are 6.6 μm and the FWHM values are 0.95 μm. The simulated results are in good agreement with the designed values. Note that there are small deviations among the simulated focus length and the designed value, which due to the approximations in selecting the radius of the resonant antenna in configuring processes and the insufficient mesh accuracy.

Then we designed three focus by the proposed metalens. According to the expression:

where the a(x) is the nearest amplitude, a₁=a₂=a₃=1/3. The designed three focus locate at x=−4,x=0, x=4 and the focal lengths are all 10 μm. We selected the radius of the resonant antenna according to Fig.7(a), which are the relationship between the position x and the radius of each unit. The Poynting vector distributions for incidence and the intensity distributions along x-direction of the simulated results are shown in Fig.7(b)and Fig.7(c). The focal lengths of Fig.7(b)are 4.7 μm and the FWHM values of Fig.7(c) are 0.86 μm. As can be seen from Fig.6 and Fig.7, the proposed structure also has a good focusing effect for multi-focus and the focal length will decrease as the focus number increases.

In summary, we proposed the metalens based on Au and MgF₂, which is independence of polarization. We simuated by using FDTD method. According to the simulated results, the proposed metalens can work well in the range of 700 nm to 850 nm and work best in the range of 750 nm to 800 nm for one focus. By configuring the resonance antenna, the proposed metalens can also work well for multi-focus. When the incidence wavelength is chosen as 800 nm, the focal length of one focus, dual focus and three focus are 9.6 μm, 6.6 μm and 4.7 μm, respectively. In conclusion, we can control the focus of the proposed metalens according to our requirements. We believe that our findings are beneficial in designing new function controlling devices.