1. Introduction

The optical lens is critically important in optical focusing and optical imaging, which has been widely used and investigated in various scientific fields. Conventional optical components depend on gradual phase accumulation to shape the transmitting wavefronts, which are achieved by either varying the refractive index or spatial distribution of the thickness of the lenses. Therefore, early traditional optical lenses make use of continuous curved surfaces to modulate the phase, which determines the bulky curved geometry of the lens and significant thickness compared to incident wavelength. However, with the development of integrated equipments, the research about ultrathin planar metalens has become an urgent requirement of integrated industry nowadays. Subsequently, Fresnel lens and plasmonic lens [1–4] are designed to reduce the volume of the lens, but the wavefront shaping still depends on the gradual phase accumulation, and the thickness of the optical lens is still in the wavelength scale at least.

In recent years, the applications of the metamaterials have attracted enormous interests due to their unusual electromagnetic properties in controlling of light propagation, such as negative refraction [5–8], invisibility cloaking [9], optical focusing [10,11], optical vortexes [12–14], and so on. Some metasurfaces consist of sub-wavelength nano-antennas which create phase shifts of 2$\pi$ for cross-polarized scattered light and posses the uniform amplitudes [15–19]. In the optical regime, advances in the field of nanofabrication and metamaterials have opened new avenues for building planar (ultrathin: sub-wavelength scale), compact optical focusing devices. In previous works, the focusing lenses have been realized for the linear polarizations by using V-shaped plasmonic nanoantenna [20–23] and nano brick [24], or for circular polarizations by using dipole nanoantennas [25], the U-shaped nanoantennas [26],and Pancharatnam–Berry phase elements [27]. However, One drawback of these focusing metalens is that they are not suitable for both circular and linear polarizations, simultaneously.

In this paper, we propose an ultra-thin broadband planar metalens by utilizing L–shaped gap antennas on a thin gold film. The designed metalens is suitable for both circular polarization (CP) and X/Y linear polarization (XLP/YLP) focusing simultaneously, which is different from previously reported metalens for focusing either linearly or circularly polarized lights only. This metasurface can generate an abrupt phase change and control the wavefront by spatially varying the phase of cross-polarization transmission light in accordance with a hyperboloidal phase profile. The phase shift from 0 to $2\pi$ can be achieved by varying two degrees of freedom, while maintain the equal amplitude. The amplitude and phase of the cross-polarization transmission light under the incidence of XLP/YLP light are the same with the case under the incidence of left and right circular polarization (LCP/RCP) light, which is decided by the specific geometric structure of our designed L–shaped gap antenna. Our metalens can work at wide incident angles and also work across a broad near-infrared spectral range. The dispersion of the designed lens has been also investigated in detail. Undoubtedly, the planar geometry of the metalens is helpful for the practical applications in integrated nano-optoelectronics and microdevices.

2. Design and methods

For generating the desired phase changes, we design a metasurface with L-shaped gap antennas. Compared with the L-shaped metal antenna, this complementary structure is favorable in reducing the background noise in the transmitted region due to blocking the incident field. The schematic of an L-shaped gap nanoantenna is shown in Fig. 1(a) , and the lengths and widths of the two arms keep the same. The $s-a$ coordinate system has an angle of 45° with respect to the $x-y$ coordinate system (as shown in Fig. 1(b)).

 

Fig. 1 (a) Schematic of the L-shaped gap antenna embedded in the thin gold film deposited on the surface of the silica glass substrate, with the cell const of $\Lambda$ = 450 nm, and gold film thickness of t = 30 nm. (b) The top view of the L-shaped gap antenna with the equal arm length L. (c) The phase shifts and amplitudes of transmitted cross-polarized light with a XLP and LCP incidence at the wavelength of 808 nm. The selected eight L-shaped gap antennas are displayed at the bottom, which correspond to different phase delays. Antennas 5-8 are rotated clockwise by 90 degrees with respect to antennas 1-4. (d) Phase shifts at different x positions of the metalens (focal length of f = 12 μm) and the corresponding L-shaped gap antennas are also depicted at the bottom of the figure.

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An arbitrary linearly polarized incident field can decomposed into two components along the $\overrightarrow{e_s}$ and $\overrightarrow{e_a}$, which will excite two eigenmodes respectively, i.e. “symmetric” and “antisymmetric” modes. The complex scattered fields of the eigenmodes ($S\overrightarrow{e_s}$and$A\overrightarrow{e_a}$) can be obtained by analytical calculations or simulations [28], where$S$and $A$ are the complex scattering amplitudes of the symmetric and antisymmetric modes. Because of the special geometric structure, under the incidences of XLP, YLP, LCP and RCP lights, the induced scattered field can be expressed as [29]:

We design a complementary L-shaped gap antenna metasurface with two DOFs of L and W in a 30 nm thick gold film to introduce abrupt phase shifts for focusing light. Finite-difference time-domain (FDTD) method is used in our numerical simulations. The phase and amplitude of the transmitted cross-polarized light can be efficiently controlled by changing the nanostructure’s length L and width W, for a fixed wavelength of 808 nm. The phase shifts are spatially arranged according to the hyperboloidal phase distributions of cylindrical lenses, and the calculated phases are quantized into eight quantization levels. Therefore, eight basic complementary L-shaped gap antennas are selected to provide the desired phase changes (from 0 to 2π with π/4 intervals) and equal intensity modulation for the cross polarized transmission light. Optical antennas with phase coverage over the whole 2$\pi$ range and equal scattering amplitudes are necessary for designing planar metalenses with a large range of focal lengths. The unit cell period Λ is 450 nm to avoid coupling between the adjacent L-shaped gap antennas. The antennas 1-4 have L = 180 nm, 170 nm, 180 nm, 150 nm; W = 30 nm, 75 nm, 125 nm, 95 nm, respectively. The antennas 5−8 can be formed by rotating clockwise antennas 1−4 90° respectively. It’s worth noting that eight basic antennas have identical amplitude and phase shift under the incidence of the XLP and LCP plane waves, as shown in Fig. 1(c). And from Fig. 1(c), we can find the amplitudes of the transmitted cross-polarized lights reach to 0.36 nearly, and therefore, the transmission of the transmitted cross-polarized lights is 13% accordingly, which is a relative larger number for the transmitted metasurface. Because of the specific symmetry of the nano-antenna in our design, changing the polarization of the incident light from XLP to YLP, or from LCP to RCP, there are no variations for the phase shift and amplitude of the transmitting lights. Therefore, the performance of the metalens should not suffer destruction due to such a change of the incident polarization, which have been successfully verified in our simulations. This indicates that the initial polarization state of the incident beam does not affect the phase and amplitude change for such an L-shaped gap antenna. These simulation results convincingly explain why our designed focusing metalens is suitable for both X/Y linear and circular polarizations simultaneously.

The basic design strategy of a planar metalens, which focuses the incident lights in the $x-z$ plane, is illustrated in Fig. 1(d). In our simulations, periodic boundary conditions are applied in the y directions, and a perfectly matched layer (PML) boundary condition are applied in the z direction. By employing the eight designed L-shaped gap antennas, a metasurface with a hyperboloidal phase profile is arranged to product the convergent transmitted wavefront. According to the Fermat’s principle, the cross polarized transmission through the metalens at different positions (different value for x) should fulfill the phase relationship: $\phi (x)=2n\pi +\frac{2\pi }{\lambda }\left(\sqrt{f^2+x^2}-f\right)$, where$\lambda$is the incident wavelength, and$f$is the designed focal length. The above equation has three variables, so if the focal length of $f$is determined, the phase shifts of different positions can be obtained. In our designed metalens, the concrete phase shifts are quantized to eight values in the range of 0 ~2π, and the eight quantized value can be obtained by our designed eight L-shaped gap antennas accordingly. Therefore, the whole designed metalens can be realized by a series of selected L-shaped gap antennas according to the concrete phase shifts at certain positions, as shown in Fig. 1(d). As for the experimental realizations, like the former reported works [6,11,19], our design can also be fabricated on a quartz glass substrate (with the refractive index of n = 1.45) with a standard electron-beam lithography, followed by metal deposition and a lift-off process.

3. Results and discussions

3.1 Focusing properties

We have designed an L–shaped metalens for focusing lights with the wavelength of 808 nm. The interface with a focal length of 12 µm is created by arranging 41 L–shaped gap antennas (Numerical aperture, NA = 0.61) according to the phase distributions of cylindrical lens. With the incidence of an XLP light, the intensity, electric field and phase distributions of the transmitted YLP light are shown in Fig. 2(a) respectively. As expected, the light is focused very well on the preset position, which is 12 µm behind the metalens. The focus size is rather small, which is in sub-wavelength scale with a FWHM (full width at half maximum) of 480 nm. The corresponding electric field distribution and wavefront of the transmitted YLP light are concave hyperboloid, so the transmitted YLP light will converge, which also means that the designed metalens is a focusing lens with a focus length of 12 µm. Due to symmetric design of the L–shaped gap unit cell, the response of metasurface lens is polarization independent. The intensities, electric fields and phase distributions with the YLP, LCP, RCP incidences are also calculated accordingly, as shown in Figs. 2(b)-2(d), and the metalens performs the same focusing phenomenons accordingly. Therefore, the designed ultra-thin nanostructure can work very well for both linear polarization (LP) and circular polarization (CP) incident lights simultaneously, that is to say that, unlike previous reported metalenses, the focusing effect of the L–shaped metalens is insensitive to the incident polarization property.

 

Fig. 2 (a)The electric field intensity, electric field and phase distributions of the transmitted YLP light indicate that the metalens is a focusing lens for XLP normal incidence, and (b-d) shows the similar cases for the YLP, LCP, RCP incidence light.

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It is worth noting that, the reported metalens based on the metasurface structures that can achieve the CP focusing, are all dual-polarity plasmonic metalens [10, 25–27], which can serve as a convex lens or concave lens according to the polarization states of the incident light. The dual-polarity effect can be attributed to the phase increment’s sign that can be reversed for different handedness of the incident light. However, here, our designed metalens is still convergent lens for both LCP and RCP incidences, because the phase increment of the L-shaped antenna is not depending on the rotation of the optical axis, and it is the same for LCP and RCP incidences.

In addition, we have also investigated the focusing behavior of the designed metalens with different incident angle (−15°, 25° 45°). As we all know, an important advantage of the planar lenses is that it can correct spherical aberration under axial (i.e. normal) illumination. However, there is off-axis aberration when the excitation is oblique incidence rather than normal incidence. As shown in Fig. 3(a) , for the XLP incident light with an angle of 15°, the electric field distributions of the transmitted YLP lights (corresponding cross-polarized lights) is concave hyperboloid and converging, and the focusing capability is maintained. The off-axis aberration of the metalens is not very obvious for oblique incidence angle of less than 15°, however, it becomes more obvious for larger oblique incident angles, which can be confirmed by the intensity distributions as shown in Fig. 3(b). Although there is large and inevitable off-axis aberration, such as astigmatism, coma and petzval field curvature, the designed metalens can still work when the incident angle is less than 45°, which implies that it is possible to achieve wide-angle focusing characteristics. When the oblique incident angle is higher than 45°, such as 50°, there is no focusing phenomenon. The distance between focus spot and axis can be expressed as$h=f\cdot \tan \left(\theta \right)$,where f is focus length, $\theta$ is the incidence angle. Figure 3 shows$h$ = 3.2, 5.2, 8.3 µm, for incidence angle of −15°, 25°, 45°, respectively, which matches very well with the theoretical distance of $h$ = 3.01, 5, 8.5 µm. That is to say aberration free focusing is possible under axial illumination for the planar lens, but diffraction-limited focusing is not possible due to off-axis aberrations under the oblique incidence with a large incidence angle .

 

Fig. 3 (a) Electric field and (b) intensity distributions of the transmitted YLP lights at oblique incidences of XLP lights. The incidence angles are −15°, 25° and 45°, respectively.

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3.2 The metalenses with different focal lengths

To further confirm the universality and validity of the designing method for metalens, two other metalenses with the different focal lengths f = 8 µm and f = 15 µm have also been designed. All of three metalenses are composed of 41 L–shaped gap antennas, therefore, in theory, when the focal lengths get longer, the NAs will become smaller and the focal intensities will get larger accordingly. The intensity distributions of the transmitted YLP light through three metalenses under the incidence of XLP light are shown in Fig. 4(a) , and the transmission lights are strongly and exactly focused at the preset positions for all of three metalenses, respectively. For a quantitative analysis of the focusing effect of three designed metalens, the cross-polarized electric field intensity along the z-axis and along the x-direction at the focal plane for the designed metalens are presented in Figs. 5(a) and 5(b) respectively. The simulated focal lengths are slightly different from the theoretical prediction, which may be due to the difference between the simple analytical formula and the sophisticated numerical model. As depicted in Fig. 5(a), the depths of focus (FWHM along the z-axis) are 1.9 μm, 3.2 μm and 4.3 μm for three designed metalens with focal lengths of 8 µm, 12 µm and 15 µm, respectively. At all focal planes, the diameters of the focal spot (FWHM along the x-axis) are in sub-wavelength scale as shown in Fig. 5(b). The corresponding FWHMs are 443 nm, 553 nm and 664 nm for three designed metalens with focal length of 8 µm, 12 µm and 15 µm, respectively, which imply that all of three designed metalens can work in higher qualities. As expected, for the metalens with shortest focal length, the shortest depth of focus and the largest transmission intensity can be achieved. The according results with the incidence of LCP light are also shown in the Fig. 4(b) and Fig. 5 for comparisons, which demonstrates that all of three designed metalenses are applicable to the LP and CP incidences very well.

 

Fig. 4 (a) Intensity distributions of the transmitted cross-polarized light (YLP) through the designed metalenses with different focal length of 8 um, 12 um, 15 um respectively, under the normal incidence of the XLP light with a wavelength of 808 nm. (b) shows the similar cases for the LCP incident light.

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Fig. 5 The intensity of the cross-polarized electric field along the z-axis (a) and along the x-axis (b) at the corresponding focusing planes with different focal lengths, under the XLP and LCP normal incidences, respectively.

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3.3 The broadband characteristics

The designed metalens can exhibit an excellent broadband focusing characteristics, which can work very well in the wavelength range of 750−1300 nm. As shown in Fig. 6(a) , the focal lengths rapidly decrease with the increase of the incident wavelengths of XLP light. As shown in Fig. 6(b), there is still the same case for the LCP incidence. The above focusing phenomenons can also be suitable for the YLP and RCP incidences. In order to precisely identify the focal length and the size of the focal point, the cross-polarized electric field intensity along the z-axis and x-direction at the focal plane for the designed metalens with different incident wavelengths for the XLP and LCP lights are also investigated and presented in Figs. 7(a) and 7(b) respectively. As expected, the strongest focus with largest intensity and smallest FWHM is achieved at the incidence wavelength of 808 nm, since the metasurface unit cell is optimal designed for this specific wavelength. Although the intensity get smaller at other incident wavelength adjacent to the 808nm, the focusing qualities are still very good as shown in Fig. 7(b), which implies that the designed metalens is indeed a broadband optical component in the near-infrared regions. As demonstrated in Fig. 7(a), the focal length decreases from 13 um to 7 um with the incident wavelength from 750 nm to 1300nm.

 

Fig. 6 (a) Intensity distributions of the YLP light through the designed metalens on the x-z plane, under the XLP normal incidence at the wavelengths of 780 nm, 808 nm, 1200 nm. (b) shows the similar cases for the LCP incidence light.

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Fig. 7 The electric field intensity along the z-axis (a) and along the x-axis (b) at the focusing plane under a series of different incident wavelengths, with the XLP and LCP normal incidence, respectively.

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The relationship between the incident wavelength and the focal lengths of the designed metalens has also been described by the red dot in Fig. 8 . For a conventional lens, based on the Fresnel approximation, phase function $\phi (x)$ can be approximated as $\phi (x)=-\frac{k}{2f}{x}^2\text{=}-\frac{\pi }{\lambda f}{x}^2$ [30]. Therefore, the relation between the focal length and incidence light wavelength can be expressed as$f_{\lambda }=\frac{{\lambda }_0}{\lambda }{f}_0$, where$f_{\lambda }$is the focal length for the actual incident wavelength λ, and $f_0$ is the preset focal length for the designed incident wavelength of ${\lambda }_0$. When λ₀ = 808 nm and f₀ = 12 µm, the chromatic aberration curve is not completely consistent with the numerical simulation results, as shown in Fig. 8. It implies that, in addition to the wavelength dispersion, the material dispersion also influence the focus shift of the metalens. The principle of our designed metalens is based on phase discontinuities of a series of unit cells. Indeed, the phase shift of the unit cell can vary significantly under different incident wavelengths.

 

Fig. 8 The comparison between the chromatic aberration curve for the conventional lens and our metalens’ focal lengths under different incident wavelengths.

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3.4 The metalenses with different NAs

The focusing ability of the lens is proportional to the NA which is defined as $NA=\sin [{\tan }^{-1}(D/2f)]$, where D is the width of the lens, and f is the focal length of the lens. For a specific focal length, NA will become larger with increasing lens' width. As shown in Fig. 9(a) , with the XLP incidences, there will be better focusing properties for the metalenses with larger NA by increasing the metalens’ width. For a quantitative analysis of the focusing effect of three metalenses (focal length of f = 12 μm), Fig. 9(c) shows that when NA = 0.76 (63 L–shaped gap antennas), the focusing intensity increases significantly, and FWHM gets smaller than the metalenses with NA = 0.50 (31 L–shaped gap antennas) and 0.61(41 L–shaped gap antennas). As well as the conventional lens, for the metalens with the larger NA, the size of the focal spot will be smaller, and the focusing field will also become stronger. As shown in Fig. 9(b), there have been the same focusing characteristics for the LCP incidence, and the focusing phenomenons above are the same as to the YLP and RCP light incidence.

 

Fig. 9 (a) Intensity distribution of the YLP light for the designed metalenses with different NA of 0.50, 0.61, 0.76 on the x-z plane, under the XLP normal incidence at a wavelength of 808 nm. (b) shows the similar cases for the LCP incidence light. (c) The electric field intensity along the x-axis at the focus plane under different incident wavelengths, with the XLP and LCP normal incidence, respectively.

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4. Conclusion

In summary, an ultra-thin, planar, broadband L–shaped metalens have been designed based on the interfacial phase discontinuity of the transmitted cross-polarized light, which is suitable for both the LP and CP incident lights focusing simultaneously. The planar design is aberration-free under the normal incidence and can also work efficiently when the incident angle less than 45°. The size of the focus spot has a superior quality and is in sub-wavelength scale. The designed metalens has a good broadband characteristics, which can work in a broad range of the incident wavelength from 750 to 1300 nm. The focal lengths will change from 13 to 7 µm accordingly, and the causes of dispersion have also been discussed and analyzed in detail. Furthermore, this metalens has the unique characteristics of polarization insensitive for the LP and CP incident lights, which is first proposed and investigated for the metalens based on phase discontinuous in the near-infrared range. It is a practical alternative to conventional lenses and quite promising for the practical application in integrated optics.