Broadband bifunctional Luneburg–Fisheye lens based on anisotropic metasurface

Luneburg lenses and Maxwell fisheye lenses possess distinct properties of focusing, well beyond conventional lenses made of uniform materials. In this paper, a planar broadband bifunctional Luneburg-fisheye lens synthesized by gradient anisotropic metasurface is proposed. The proposed anisotropic metasurface is formed by non-resonant anisotropic cells, so that it can independently realize the equivalent gradient refractive indexes of Luneburg lens and Maxwell fisheye lens along orthogonal directions in a broad band, respectively. To verify the performance of the bifunctional lens, a prototype associated with a feeding log-periodic dipole antenna has been fabricated. Experimental results show that the proposed lens functions well over a wide frequency range with high efficiency and low profile, which coincides well with theoretical predictions and simulated results. It is expected that the proposed design will facilitate the applications of multifunctional metadevices in microwave and optical ranges.

In this paper, a broadband bifunctional Luneburg-fisheye lens constructed by gradient anisotropic metasurface (GAM) is proposed, which characters as a Luneburg lens viewing from the horizontal axis in the frequency range of 2 to 8 GHz (relative bandwidth of 120%), while as a Maxwell fisheye lens viewing from the vertical axis in the band of 5 to 7 GHz (relative bandwidth of 33.3%). The proposed design of the Luneburg-fisheye lens stands out from the aforementioned designs in three aspects. First, the proposed metalens can achieve a bifunctional response of a Luneburg lens and a fisheye lens over a broad band far beyond the existing designs. The constitutive GAM is synthesized by non-resonant anisotropic cells applying with quasi-conformal mapping method, so that it can independently regulate the inhomogeneous refractive-index distributions in two orthogonal directions. Second, the bifunctional lens is designed to be flattened with compact size and low loss, resulting in easy integration with other devices. Third, what needs to be highlighted is that this anisotropy-based approach is not limited to a combination of the two lenses. It can be flexibly extended to the realization of other broadband bifunctional metadevices, such as Luneburg-Eaton lens, Eaton-fisheye lens, and so on.

Results
Design and analysis of the anisotropic unit cell. The configuration of the proposed anisotropic unit cell used for the synthesis of the bifunctional lens is demonstrated in Fig. 1. The anisotropic cell consists of two mutually orthogonal parallel-line arrays. The two sets of the metallic array are attached separately on both sides of a commercial F4B substrate (ε r = 2.2, tan δ = 0.001) with a thickness of t = 1 mm (0.027λ at 8 GHz). Considering the operating band of the metalens, the periodicity p of the anisotropic cell is set as 6 mm.
As mentioned above, it is extremely difficult to realize a bifunctional lens with isotropic materials. Here, we will discuss the feasibility of realizing a Luneburg-fisheye lens with anisotropic metasurface (AM). Suppose that a point source is located on xoy-plane to illuminate the synthesized metalens, which is inhomogeneous, as shown in Fig. 2. When the EM wave emitted from the point source travels along x axis, the refractive index of the cell block is denoted as n x (like Unit A). Similarly, when the EM wave propagates along y axis, the refractive index of the cell block is denoted as n y . In the case of oblique incidence with an angle of θ relative to the x-axis, we rotate the former coordinates counterclockwise by the angle of θ so that a new coordinate system can be formed, marked by x'-y' axes, as depicted in Unit B. The coordinate transformation resulting from the rotation can be expressed as follows: With respect to the new coordinate system, the refractive index tensor of the cell block can be calculated by the following expression.
As we know, to realize a bifunctional metalens, the metasurface should be anisotropic in orthogonal directions (i.e. x-and y-axis). While, the AM is also expected to remain nearly isotropic along each axis under the circumstance of oblique incidence. As indicated by expression (2), when the incident angle θ is small, n x'x' and n y'y' is approximately equal to n x and n y , respectively. Furthermore, the cross-polarization components n x'y' and n y'x' are approaching zero in this case. However, if θ is getting larger, the preceding conclusions will only be applicable when the values of n x and n y are close to each other. According to the previous descriptions of the Luneburg lens n x cos 2 θ + n y sin 2 θ n y ′ y ′ = n x sin 2 θ + n y cos 2 θ n x ′ y ′ = n y ′ x ′ = (n y − n x ) sin θ cos θ   [1,2], respectively. Hence, the farther away from the center of the lens, which is corresponding to the increasing of θ, the closer the values of the refractive index will be in orthogonal directions. In other words, when θ is large, the deviation value between n x and n y could be very small. Thus, n x'x' and n y'y' can still approximate to n x and n y , respectively, and the cross couplings can also be diminished (i.e. n x'y' ≈ n y'x' ≈ 0). In brief, the values of n x and n y can be modulated independently for small incident angles; whereas the values of n x and n y should be designed approximately identical for the situations of large-angle incidence. From the analyses above, it is feasible to fulfill the requirements of a Luneburg-fisheye lens with an AM theoretically, applying the quasi-conformal mapping method 29,36 .
To further validate the anisotropy of the proposed unit cell, the characteristics of the anisotropic cell are simulated by commercial software of CST Microwave Studio Suite. Provided that x-polarized incident waves propagate along y-axis as shown in Fig. 1, the variations of transmission and reflection coefficient versus frequency are plotted in Fig. 3. As can be seen from Fig. 3, despite changing the geometrical dimensions of the bottom structure arbitrarily, the differences between the values of S 11 or S 21 could be tiny, as long as the top structure remains unchanged (i.e. l 1 , d 1 , and g 1 are fixed). Conversely, similar results will be achieved, given that the structure of the top and the bottom metallic layers is similar but oriented orthogonally. Obviously, the proposed unit cell can be anisotropic and the EM responses along x-and y-axis can be regulated independently as well. Moreover, magnitudes of the transmission coefficient S 21 are all close to 0 in Fig. 3, indicating high transmission of the proposed AM (≥ 0.9).  Frequency (GHz) Arg(S 21 ) for l 2 =0mm,d 2 =0mm,g 2 =0mm www.nature.com/scientificreports/ Applying effective medium theory, the effective refractive index of the proposed anisotropic cells can be retrieved by a S-parameter inversion method 37,38,39 . According to the correlative reflection and transmission coefficient shown in Fig. 3, the variations of effective refractive index for different bottom structural dimensions are displayed in Fig. 4. If the dimensions of top structure remain invariable, the effect on the retrieved parameters will be negligible, regardless of the sizes of the bottom structure, and the values of the effective refractive index are nearly constant in a wide frequency range as expected. In addition, the imaginary part of the extracted refractive index is roughly equal to 0, signifying low loss of the proposed lens.
Another difficulty of realizing a broadband Luneburg-fisheye lens lies in the coverage of the effective refractive index. As can be seen from Fig. 3, the proposed anisotropic cell operates in a non-resonant region, leading to an inherently broadband property and low loss. Since the structure of the top layer is similar to that of the bottom layer, and the refractive index distributions in orthogonal directions can be adjusted independently, the refractive indexes of the anisotropic cells with varying structural parameters are extracted when the values of l 2 , d 2 , and g 2 are fixed, as shown in Fig. 5. The variations of the effective refractive indexes versus frequency are minor, and by varying the geometries of the proposed anisotropic cell, the requirements of the refractive index distribution for the two lenses can be achieved in a wide band. The detailed variations of structural parameters are listed in Table 1. Accordingly, the corresponding values of the effective refractive index n x for Luneburg lens can range from 1.04 to 1.40 in the band of 2 to 8 GHz, while, that of n y for Maxwell fisheye lens can range from 1.04 to 1.99 in the band of 5 to 7 GHz.
Design of the bifunctional Luneburg-fisheye lens. Sequel to the characteristics of the anisotropic cell, a broadband Luneburg-fisheye lens can be synthesized based on the specified refractive-index distributions of the two lenses.
The refractive index distribution of the Luneburg lens satisfies the following function of spatial position:   where R is the radius of the lens, as shown in Fig. 6, and r is the radial distance away from the center of the lens. The refractive index distribution of the Maxwell fisheye lens obeys:    www.nature.com/scientificreports/ Here, R is the radius of the lens, and r is the radial coordinate as well. n 0 represents the maximum refractive index at the center of the lens, whose value is set as 2 generally.
As demonstrated in Fig. 6, stepped refractive index profiles are employed to approximate the theoretical ones calculated by (3) and (4). In order to integrate the functions of the two lenses into a single dielectric slab, an anisotropic metasurface consists of 20 × 20 cell blocks is designed, whose overall size is 120 mm × 120 mm × 1 mm (i.e. 3.2λ × 3.2λ × 0.027λ at 8 GHz). The discretized refractive-index distributions of the two lenses are displayed (4) n y (r) = n 0 [1 + (r R) 2 ], (0 ≤ r ≤ R)  www.nature.com/scientificreports/ in Fig. 7, based on a method of quasi-conformal mapping. For practical implementation, the Luneburg lens and the fisheye lens is finally discretized into seven and ten concentric circular regions, respectively.
Once the dimensions of the filling cells are ascertained in accordance with the refractive index distributions arranged in Fig. 7, a broadband bifunctional Luneburg-fisheye lens can be realized, as shown in Fig. 8. It can be seen clearly that the synthesized AM behaves as a Luneburg lens viewing from the horizontal orientation, while behaves as a Maxwell fisheye lens viewing from the vertical orientation. The metasurface on the top layer corresponds to the Luneburg lens (blue part), and the one on the bottom layer corresponds to the fisheye lens (pink part). Under the illumination of orthogonally polarized waves, the two lenses can achieve their distinct functions of beam focusing independently (see the blue and pink ray tracing, respectively).  Fig. 9a. With which, the bifunctional Luneburg-fisheye lens can be excited by directional TE-mode surface waves, fulfilling the requirement of the proposed AM. The LPDA is printed on a commercial FR4 substrate (ε r = 4.4, and tan δ = 0.02) with a thickness of 1 mm. The impedance characteristic (S 11 ) and the radiation performance of the LPDA is detailedly presented in Fig. 9, respectively. As shown, the designed LPDA roughly operates over a wide frequency range from 2 to 8 GHz, and exhibits a desired performance of directional radiation along the central axis (see Fig. 9b-d).
Aiming to improve the performance of the proposed bifunctional lens, a triple-layered structure is adopted, and the stacking AMs are spaced 6 mm apart from each other. The schematic diagram of the whole system after assembly is demonstrated in Fig. 10. Specifically, the feeding LPDA is coplanar with the middle AM, locating on the periphery of the metalens, and the central axis of the LPDA is aligned to that of the middle layer.
Finally, to verify the performance of the proposed bifunctional metalens, a triple-layered prototype associated with the feeding LPDA is fabricated, assembled, and measured, as shown in Fig. 11. The three pieces of AM are sandwiched by foam spacers, whose relative permittivity and relative permeability are close to 1. The surroundings of far-field and near-field measurement are displayed in Fig. 11b, c, respectively. Besides, the scanning area of near field is 420 mm × 300 mm, which is sufficient to cover the entire lens system (marked by yellow rectangle), and the components of the tangential electrical fields on the metasurface are collected by a waveguide probe.
As described in Fig. 8, the proposed bifunctional metalens will function as a broadband Luneburg lens when the anisotropic metasurfaces are illuminated from y axis. The measured distributions of the instantaneous near electric field on the lens surface (xoy-plane) at 4, 6, and 8 GHz are compared with the numerical simulations successively, as demonstrated in Fig. 12. As can be seen from Fig. 12a-c, the measured electric-field distributions are in great agreement with the simulated ones, and the spherical waves radiated from the LPDA are dramatically transformed into quasi-plane waves, as predicted. To further reveal the performance of the metalens, the phase distributions of electric-field on xoy-plane are presented in Fig. 12d,e,f, as well. Likewise, it is obvious that the spherical wavefront can be converted to the quasi-planar wavefront in a broad band, and the effect of conversion is enhanced with the increase of frequency. For quantitative demonstration, simulated and measured gains versus frequency of the assembled tri-layered lens antenna (TLLA) and the feeding LPDA are plotted in Fig. 13. A significant gain enhancement can be observed from 2 to 8 GHz (relative bandwidth of 120%), which indicates a highly directional emission resulting from the Luneburg lens.
Attributing to the anisotropy of the metasurface, the bifunctional metalens will function as a wide-band Maxwell fisheye lens when we rotate it 90 degrees clockwise with the location of the LPDA fixed. The comparisons of the instantaneous near electric fields on xoy-plane at 5, 6, and 7 GHz are demonstrated in Fig. 14. The experimental measurements agree well with the numerical simulations. Both amplitude and phase distributions of electric field show that the radiation emitted from the LPDA is focused on the opposite periphery of the metalens, and then propagates as spherical waves in a band of 5 to 7 GHz (relative bandwidth of 33.3%).
In order to investigate the influence of cross-polarized electric fields on the performance of the bifunctional metalens, the simulated cross-polarized distributions of electric-field at 6 GHz for the Luneburg lens (E y ) and Maxwell fisheye lens (E x ) are illustrated in Fig. 15a,b, respectively. With respect to the TE-mode surface waves, the Poynting vector can be calculated as expression (5). According to the mechanism of the bifunctional metalens, the electric field E x plays a dominant role in exciting the Luneburg lens, which is designed along y-axis. Similarly, the electric field E y plays a dominant role in exciting the fisheye lens, which is designed along x-axis. Hence, it can be seen that the electric-field components of cross-polarization on the lens surface depicted in Fig. 15a,b are quite smaller than the corresponding components of co-polarization shown in Figs. 12 and 14. Moreover, compared with the theoretical predictions of beam path in Fig. 8, the power-density distributions in Fig. 15c,d show that the cross-polarized components of electric field do not much affect the power flows of the Luneburg and fisheye lenses as expected.

Conclusions
A broadband bifunctional Luneburg-fisheye lens based on anisotropic metasurface is proposed. The planar bifunctional metalens is synthesized by non-resonant anisotropic cells with low profile, so that it can achieve distinctly inhomogeneous distributions of refractive index in orthogonal directions, respectively. By virtue of the anisotropy, the bifunctional metalens can serve as a broadband Luneburg lens observing from the horizontal www.nature.com/scientificreports/ axis in the frequency range of 2 to 8 GHz, while as a wide-band Maxwell fisheye lens observing from the vertical axis in the operating band of 5 to 7 GHz. What's more, our anisotropy-based approach is not only limited to an implementation of a Luneburg-fisheye lens, but can be flexibly applied to the realization of other broadband multifunctional metadevices operating in the region of microwave, optics, and even terahertz.